86:"UserProgressProvider" 87:I[651451,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$50","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d"],"$86"] 88:I[196793,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$50","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d"],"Gingerbread"] 8a:"TopicRecentlyViewed" 8b:I[716297,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$50","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d"],"$8a"] 8c:"RecordResourceResume" 8d:I[647935,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$50","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d"],"$8c"] 8e:"TopicQuestionBank" 8f:I[426249,["$1","$2","$3","$4","$5","$6","$7","$8","$9","$a","$b","$c","$d","$e","$f","$10","$11","$12","$13","$19","$1a","$1b","$1c","$1d","$1e","$1f","$20","$21","$22","$23","$24","$25","$26","$27","$28","$50","$51","$52","$53","$54","$55","$56","$57","$58","$59","$5a","$5b","$5c","$5d","$5e","$5f","$60","$61","$62","$63","$64","$65","$66","$67","$68","$69","$6a","$6b","$6c","$6d"],"$8e"] 90:T406,Next, we calculate the $x$-coordinates from $x_0 = 0$ to $x_8 = 2$ in steps of $h = 0.25$, and evaluate our integrand function, $f(x) = \frac{10}{1 + e^{-x}}$, at each point. Let's construct a table of values (rounding to 3 decimal places for our working out): $$\begin{aligned} x_0 &= 0 & y_0 &= f(0) = \frac{10}{1 + e^{0}} = 5 \\ x_1 &= 0.25 & y_1 &= f(0.25) = \frac{10}{1 + e^{-0.25}} \approx 5.622 \\ x_2 &= 0.5 & y_2 &= f(0.5) = \frac{10}{1 + e^{-0.5}} \approx 6.225 \\ x_3 &= 0.75 & y_3 &= f(0.75) = \frac{10}{1 + e^{-0.75}} \approx 6.792 \\ x_4 &= 1 & y_4 &= f(1) = \frac{10}{1 + e^{-1}} \approx 7.311 \\ x_5 &= 1.25 & y_5 &= f(1.25) = \frac{10}{1 + e^{-1.25}} \approx 7.773 \\ x_6 &= 1.5 & y_6 &= f(1.5) = \frac{10}{1 + e^{-1.5}} \approx 8.176 \\ x_7 &= 1.75 & y_7 &= f(1.75) = \frac{10}{1 + e^{-1.75}} \approx 8.520 \\ x_8 &= 2 & y_8 &= f(2) = \frac{10}{1 + e^{-2}} \approx 8.808 \end{aligned}$$ Identifying the correct $x$-values earns **(M1)**, and finding the correct $y$-values earns another method mark **(M1)**.91:T406,Next, we calculate the $x$-coordinates from $x_0 = 0$ to $x_8 = 2$ in steps of $h = 0.25$, and evaluate our integrand function, $f(x) = \frac{10}{1 + e^{-x}}$, at each point. Let's construct a table of values (rounding to 3 decimal places for our working out): $$\begin{aligned} x_0 &= 0 & y_0 &= f(0) = \frac{10}{1 + e^{0}} = 5 \\ x_1 &= 0.25 & y_1 &= f(0.25) = \frac{10}{1 + e^{-0.25}} \approx 5.622 \\ x_2 &= 0.5 & y_2 &= f(0.5) = \frac{10}{1 + e^{-0.5}} \approx 6.225 \\ x_3 &= 0.75 & y_3 &= f(0.75) = \frac{10}{1 + e^{-0.75}} \approx 6.792 \\ x_4 &= 1 & y_4 &= f(1) = \frac{10}{1 + e^{-1}} \approx 7.311 \\ x_5 &= 1.25 & y_5 &= f(1.25) = \frac{10}{1 + e^{-1.25}} \approx 7.773 \\ x_6 &= 1.5 & y_6 &= f(1.5) = \frac{10}{1 + e^{-1.5}} \approx 8.176 \\ x_7 &= 1.75 & y_7 &= f(1.75) = \frac{10}{1 + e^{-1.75}} \approx 8.520 \\ x_8 &= 2 & y_8 &= f(2) = \frac{10}{1 + e^{-2}} \approx 8.808 \end{aligned}$$ Identifying the correct $x$-values earns **(M1)**, and finding the correct $y$-values earns another method mark **(M1)**.71:["$","$L87",null,{"bundle":{"topicIds":["1575b9fa-8d09-465b-8f78-79c2e5c1591b"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L88",null,{"feature":"question-bank"}],"$L89",["$","$L8b",null,{"subjectCode":"a_level_maths","subjectName":"Maths","topicSlug":"a-level-pure-mathematics-11-9-the-trapezium-rule","topicTitle":"11.9 The Trapezium Rule","sectionTitle":null,"activity":"worksheet","resourceTitle":"11.9 The Trapezium Rule Questions","resourceHref":"/a-level/maths/edexcel/a-level-pure-mathematics-11-9-the-trapezium-rule/questions/q1","saveTrail":false}],["$","$L8d",null,{"href":"/a-level/maths/edexcel/a-level-pure-mathematics-11-9-the-trapezium-rule/questions/q1","label":"11.9 The Trapezium Rule","subjectCode":"a_level_maths","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 11 – 11.9 The Trapezium Rule - 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0}{8} = 0.25$ M1\n- Evaluate function at $x = 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2$ M1\n- Values: $5, 5.622, 6.225, 6.792, 7.311, 7.773, 8.176, 8.520, 8.808$ M1\n- Apply formula: $0.25 \\left( \\frac{5}{2} + 5.622 + 6.225 + 6.792 + 7.311 + 7.773 + 8.176 + 8.520 + \\frac{8.808}{2} \\right)$ M2\n- Final answer $14.3$ (3sf) A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Calculates the correct step size h = 0.25","code":"M"},{"id":"p2","marks":1,"description":"Lists the correct x-values from 0 to 2 in increments of 0.25","code":"M"},{"id":"p3","marks":1,"description":"Calculates corresponding y-values correctly","code":"M"},{"id":"p4","marks":2,"description":"Correct application of the trapezium rule formula structure","code":"M"},{"id":"p5","marks":1,"description":"Provides the final answer 14.3 correctly rounded to 3 significant figures","code":"A"}],"finalAnswers":["14.3"],"generatedAt":"2026-05-06T14:05:09.732Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"trapeziumformula","label":"Standard Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the step size","content":"To begin, we need to calculate the width of each strip, $h$, using the interval boundaries and the given number of strips. \n\nThe formula for the step size is:\n\n$$h = \\frac{b - a}{n}$$\n\nWhere:\n- $a = 0$ (lower limit of integration)\n- $b = 2$ (upper limit of integration)\n- $n = 8$ (number of strips)\n\nSubstituting these values in, we get:\n\n$$h = \\frac{2 - 0}{8} = 0.25$$\n\nCalculating this step size correctly earns you the first method mark **(M1)**.","highlights":[{"text":"8 strips","location":"specification"},{"text":"0","location":"specification"},{"text":"2","location":"specification"}]},{"type":"text","order":1,"title":"Evaluate the integrand values","content":"$90","calculator":[{"gdc":{"expression":"y = 10 / (1 + e^(-x))","instructions":"1. Go to the Table menu.\n2. Input the function Y1 = 10 / (1 + e^(-x)).\n3. Set the table range: Start = 0, End = 2, Step = 0.25.\n4. Display the table to read the corresponding y-values."},"ti84":{"expression":"Y1 = 10 / (1 + e^(-X))","instructions":"1. Press [MODE] and ensure 'NORMAL' is selected.\n2. Press [Y=] and enter Y1 = 10 / (1 + e^(-X)).\n3. Press [2nd] [WINDOW] (TBLSET) and set TblStart = 0 and ΔTbl = 0.25.\n4. Press [2nd] [GRAPH] (TABLE) to view the calculated values of Y1."},"desmos":{"expression":"y_1 = 10 / (1 + e^(-x_1))","instructions":"1. Click the '+' button and select 'Table'.\n2. In the first column header (x1), enter the values: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2.\n3. In the second column header, type '10 / (1 + e^(-x1))'.\n4. The calculated y-values will instantly populate the table."},"description":"Generate a table of values directly using the Table feature on your graphing calculator."}]},{"type":"hyper_latex","block":{"nodes":[{"text":"The standard trapezium rule formula is given by:"},{"latex":"$$$\\text{Area} \\approx \\frac{\\htmlData{anchor=step-h}{\\color{@sky}{h}}}{2} \\left[ \\htmlData{anchor=ends}{\\color{@amber}{y_0 + y_n}} + 2(\\htmlData{anchor=middle}{\\color{@purple}{y_1 + y_2 + \\dots + y_{n-1}}}) \\right]$$"},{"text":"We can rewrite this equivalent form to clearly show the boundaries and the internal ordinates:"},{"latex":"$$$\\text{Area} \\approx \\htmlData{anchor=step-sub}{\\color{@sky}{h}} \\left[ \\frac{\\htmlData{anchor=y0-sub}{\\color{@amber}{y_0}}}{2} + y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 + \\frac{\\htmlData{anchor=yn-sub}{\\color{@amber}{y_8}}}{2} \\right]$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"step-h"},{"color":"amber","style":"text-color","anchor":"ends"},{"color":"purple","style":"text-color","anchor":"middle"}],"connections":[{"to":"step-sub","from":"step-h","color":"sky","label":null,"style":"solid"},{"to":"y0-sub","from":"ends","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Apply standard trapezium formula","description":"This is the general formula. Let's substitute our values directly into it next."},{"type":"text","order":3,"title":"Substitute the calculated values","content":"Now we substitute our values into the formula. Showing this substitution step is vital to secure **(M2)** marks on the exam:\n\n$$\\text{Area} \\approx 0.25 \\left( \\frac{5}{2} + 5.622 + 6.225 + 6.792 + 7.311 + 7.773 + 8.176 + 8.520 + \\frac{8.808}{2} \\right)$$\n\nFirst, let's sum the interior values and the halved boundary values:\n\n- Halved end-points: $\\frac{5}{2} + \\frac{8.808}{2} = 2.5 + 4.404 = 6.904$\n- Interior sum: $5.622 + 6.225 + 6.792 + 7.311 + 7.773 + 8.176 + 8.520 = 50.419$\n\n$$\\text{Total Sum} = 6.904 + 50.419 = 57.323$$\n\nNow multiply by the step size $h = 0.25$:\n\n$$\\text{Area} \\approx 0.25 \\times 57.323 = 14.33075$$"},{"type":"text","order":4,"title":"State the final answer","content":"Finally, we round our calculated value to 3 significant figures, as is standard in examinations unless stated otherwise.\n\n$$\\text{Area} \\approx 14.3 \\text{ (3sf)}$$\n\nThis final rounded value secures the final accuracy mark **(A1)**."}]}],"generatedAt":"2026-06-11T03:16:18.643Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b307692b-a841-4dd1-a795-903d277233bf:trapeziumformula","label":"Standard Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the step size","content":"To begin, we need to calculate the width of each strip, $h$, using the interval boundaries and the given number of strips. \n\nThe formula for the step size is:\n\n$$h = \\frac{b - a}{n}$$\n\nWhere:\n- $a = 0$ (lower limit of integration)\n- $b = 2$ (upper limit of integration)\n- $n = 8$ (number of strips)\n\nSubstituting these values in, we get:\n\n$$h = \\frac{2 - 0}{8} = 0.25$$\n\nCalculating this step size correctly earns you the first method mark **(M1)**.","highlights":[{"text":"8 strips","location":"specification"},{"text":"0","location":"specification"},{"text":"2","location":"specification"}]},{"type":"text","order":1,"title":"Evaluate the integrand values","content":"$91","calculator":[{"gdc":{"expression":"y = 10 / (1 + e^(-x))","instructions":"1. Go to the Table menu.\n2. Input the function Y1 = 10 / (1 + e^(-x)).\n3. Set the table range: Start = 0, End = 2, Step = 0.25.\n4. Display the table to read the corresponding y-values."},"ti84":{"expression":"Y1 = 10 / (1 + e^(-X))","instructions":"1. Press [MODE] and ensure 'NORMAL' is selected.\n2. Press [Y=] and enter Y1 = 10 / (1 + e^(-X)).\n3. Press [2nd] [WINDOW] (TBLSET) and set TblStart = 0 and ΔTbl = 0.25.\n4. Press [2nd] [GRAPH] (TABLE) to view the calculated values of Y1."},"desmos":{"expression":"y_1 = 10 / (1 + e^(-x_1))","instructions":"1. Click the '+' button and select 'Table'.\n2. In the first column header (x1), enter the values: 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2.\n3. In the second column header, type '10 / (1 + e^(-x1))'.\n4. The calculated y-values will instantly populate the table."},"description":"Generate a table of values directly using the Table feature on your graphing calculator."}]},{"type":"hyper_latex","block":{"nodes":[{"text":"The standard trapezium rule formula is given by:"},{"latex":"$$$\\text{Area} \\approx \\frac{\\htmlData{anchor=step-h}{\\color{@sky}{h}}}{2} \\left[ \\htmlData{anchor=ends}{\\color{@amber}{y_0 + y_n}} + 2(\\htmlData{anchor=middle}{\\color{@purple}{y_1 + y_2 + \\dots + y_{n-1}}}) \\right]$$"},{"text":"We can rewrite this equivalent form to clearly show the boundaries and the internal ordinates:"},{"latex":"$$$\\text{Area} \\approx \\htmlData{anchor=step-sub}{\\color{@sky}{h}} \\left[ \\frac{\\htmlData{anchor=y0-sub}{\\color{@amber}{y_0}}}{2} + y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 + \\frac{\\htmlData{anchor=yn-sub}{\\color{@amber}{y_8}}}{2} \\right]$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"step-h"},{"color":"amber","style":"text-color","anchor":"ends"},{"color":"purple","style":"text-color","anchor":"middle"}],"connections":[{"to":"step-sub","from":"step-h","color":"sky","label":null,"style":"solid"},{"to":"y0-sub","from":"ends","color":"amber","label":null,"style":"solid"}]},"order":2,"title":"Apply standard trapezium formula","description":"This is the general formula. Let's substitute our values directly into it next."},{"type":"text","order":3,"title":"Substitute the calculated values","content":"Now we substitute our values into the formula. Showing this substitution step is vital to secure **(M2)** marks on the exam:\n\n$$\\text{Area} \\approx 0.25 \\left( \\frac{5}{2} + 5.622 + 6.225 + 6.792 + 7.311 + 7.773 + 8.176 + 8.520 + \\frac{8.808}{2} \\right)$$\n\nFirst, let's sum the interior values and the halved boundary values:\n\n- Halved end-points: $\\frac{5}{2} + \\frac{8.808}{2} = 2.5 + 4.404 = 6.904$\n- Interior sum: $5.622 + 6.225 + 6.792 + 7.311 + 7.773 + 8.176 + 8.520 = 50.419$\n\n$$\\text{Total Sum} = 6.904 + 50.419 = 57.323$$\n\nNow multiply by the step size $h = 0.25$:\n\n$$\\text{Area} \\approx 0.25 \\times 57.323 = 14.33075$$"},{"type":"text","order":4,"title":"State the final answer","content":"Finally, we round our calculated value to 3 significant figures, as is standard in examinations unless stated otherwise.\n\n$$\\text{Area} \\approx 14.3 \\text{ (3sf)}$$\n\nThis final rounded value secures the final accuracy mark **(A1)**."}]}],"generatedAt":"2026-06-11T03:16:18.712Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"2469c899-7969-4983-932e-a0d7e5a414ee","revisionId":"be95ca57-fef5-41ba-a5d3-8365d53f5586","questionNumber":"3","questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":"11.9 The Trapezium Rule","calculatorUse":null,"isCalculation":true,"parts":[{"id":"1934f05f-e699-4096-b6ae-0b1793258fd1","label":"a","content":"Use the trapezium rule with 8 strips to estimate the value of\n\n$$\\int_{-2}^2 e^{x^2} \\, dx$$","metadata":null,"markscheme":"- Calculate step size $h = \\frac{2 - (-2)}{8} = 0.5$ M1\n- Evaluate function at $x = -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2$ M1\n- Values: $e^4, e^{2.25}, e^1, e^{0.25}, 1, e^{0.25}, e^1, e^{2.25}, e^4$ M1\n- Apply formula: $0.5 \\left( \\frac{e^4}{2} + e^{2.25} + e + e^{0.25} + 1 + e^{0.25} + e + e^{2.25} + \\frac{e^4}{2} \\right)$ M2\n- Final answer $41.3$ (3sf) A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Correctly identifies the interval width $h = 0.5$.","code":"M"},{"id":"p2","marks":2,"description":"Correctly evaluates the function at the required $x$ values.","code":"M"},{"id":"p3","marks":2,"description":"Correct substitution into the trapezium rule formula.","code":"M"},{"id":"p4","marks":1,"description":"Final answer of $41.3$ (3sf).","code":"A"}],"finalAnswers":["41.3"],"generatedAt":"2026-05-06T03:41:08.106Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_trapezium","label":"Direct Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the step size","content":"The trapezium rule is used to estimate the area under a curve by dividing it into a series of strips. First, we need to calculate the width of each strip, known as the step size $h$.\\n\\nThe formula for the step size is:\\n\\n$$h = \\frac{b - a}{n}$$\\n\\nwhere:\\n- $a = -2$ (the lower limit of integration)\\n- $b = 2$ (the upper limit of integration)\\n- $n = 8$ (the number of strips)\\n\\nSubstituting these values in gives:\\n\\n$$h = \\frac{2 - (-2)}{8} = \\frac{4}{8} = 0.5$$"},{"type":"text","order":1,"title":"Identify the x-values","content":"Using our step size of $h = 0.5$, we list all the $x$-coordinates starting from the lower limit $x_0 = -2$ up to the upper limit $x_8 = 2$:\\n\\n$$\\begin{aligned} x_0 &= -2 \\\\ x_1 &= -1.5 \\\\ x_2 &= -1 \\\\ x_3 &= -0.5 \\\\ x_4 &= 0 \\\\ x_5 &= 0.5 \\\\ x_6 &= 1 \\\\ x_7 &= 1.5 \\\\ x_8 &= 2 \\end{aligned}$$"},{"type":"text","order":2,"title":"Evaluate the function","content":"Now, we evaluate our function $f(x) = e^{x^2}$ at each of these $x$-values. It is best to keep these in exact exponential form to avoid rounding errors during the intermediate steps:\\n\\n$$\\begin{aligned} y_0 &= f(-2) = e^{(-2)^2} = e^4 \\\\ y_1 &= f(-1.5) = e^{(-1.5)^2} = e^{2.25} \\\\ y_2 &= f(-1) = e^{(-1)^2} = e^1 \\\\ y_3 &= f(-0.5) = e^{(-0.5)^2} = e^{0.25} \\\\ y_4 &= f(0) = e^{0^2} = e^0 = 1 \\\\ y_5 &= f(0.5) = e^{0.5^2} = e^{0.25} \\\\ y_6 &= f(1) = e^{1^2} = e^1 \\\\ y_7 &= f(1.5) = e^{1.5^2} = e^{2.25} \\\\ y_8 &= f(2) = e^{2^2} = e^4 \\end{aligned}$$"},{"type":"text","order":3,"title":"Apply trapezium rule formula","content":"The general formula for the trapezium rule is:\\n\\n$$\\int_{a}^{b} f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\\n\\nAlternatively, we can write this directly as a sum of individual trapeziums:\\n\\n$$\\int_{-2}^{2} e^{x^2} \\, dx \\approx h \\left( \\frac{y_0}{2} + y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 + \\frac{y_8}{2} \\right)$$\\n\\nSubstituting our values into this formula:\\n\\n$$\\text{Estimate} = 0.5 \\left( \\frac{e^4}{2} + e^{2.25} + e^1 + e^{0.25} + 1 + e^{0.25} + e^1 + e^{2.25} + \\frac{e^4}{2} \\right)$$"},{"type":"text","order":4,"title":"Calculate the final estimate","content":"Combining the matching terms simplifies our calculation:\\n\\n$$\\text{Estimate} = 0.5 \\left[ e^4 + 2e^{2.25} + 2e + 2e^{0.25} + 1 \\right]$$\\n\\nNow, we calculate the numerical values using a scientific calculator:\\n- $e^4 \\approx 54.5982$\\n- $2e^{2.25} \\approx 2 \\times 9.4877 = 18.9755$\\n- $2e^1 \\approx 2 \\times 2.7183 = 5.4366$\\n- $2e^{0.25} \\approx 2 \\times 1.2840 = 2.5681$\\n- $1$\\n\\nSumming these values:\\n\\n$$54.5982 + 18.9755 + 5.4366 + 2.5681 + 1 = 82.5784$$\\n\\nFinally, multiply by $0.5$:\\n\\n$$\\text{Estimate} = 0.5 \\times 82.5784 = 41.2892$$\\n\\nRounding to $3$ significant figures, we get:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx \\approx 41.3$$","calculator":[{"gdc":{"expression":"0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)","instructions":"Enter the expression into the run-matrix menu:\n0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)"},"ti84":{"expression":"0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)","instructions":"Type the simplified sum directly on the home screen:\nPress [0.5] [*(] [2nd] [LN] (e^x) [4] [)] [+] [2] [2nd] [LN] [2.25] [)] [+] [2] [2nd] [LN] [1] [)] [+] [2] [2nd] [LN] [0.25] [)] [+] [1] [Enter]."},"desmos":{"expression":"0.5 * (e^{4} + 2e^{2.25} + 2e^{1} + 2e^{0.25} + 1)","instructions":"Type the direct expression into a cell to get the calculated decimal value."},"description":"Evaluate the definite integral approximation directly on a graphing calculator or tool."}]}]},{"id":"symmetry_trapezium","label":"Using Symmetry","steps":[{"type":"text","order":0,"title":"Exploit function symmetry","content":"The integrand function $f(x) = e^{x^2}$ is an **even function**, because:\\n\\n$$f(-x) = e^{(-x)^2} = e^{x^2} = f(x)$$\\n\\nDue to this symmetry about the $y$-axis, the area under the curve from $-2$ to $2$ is exactly double the area from $0$ to $2$:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx = 2 \\int_{0}^2 e^{x^2} \\, dx$$\\n\\nInstead of applying the trapezium rule with 8 strips over $[-2, 2]$, we can apply it with 4 strips over $[0, 2]$ and then multiply our final estimate by 2. This significantly reduces our calculation steps and minimizes errors."},{"type":"text","order":1,"title":"Calculate step size","content":"For 4 strips over the interval $[0, 2]$, we calculate the step size $h$ using the formula:\\n\\n$$h = \\frac{b - a}{n}$$\\n\\nSubstituting our values:\\n\\n$$h = \\frac{2 - 0}{4} = 0.5$$"},{"type":"text","order":2,"title":"Create coordinate table","content":"Starting at $x_0 = 0$ and adding $h = 0.5$ at each step, we find the coordinates and evaluate $y = e^{x^2}$:\\n\\n$$\\begin{array}{|c|l|l|} \\hline i & x_i & y_i = e^{x_i^2} \\\\ \\hline 0 & 0 & e^{0^2} = 1 \\\\ 1 & 0.5 & e^{0.5^2} = e^{0.25} \\approx 1.284025 \\\\ 2 & 1.0 & e^{1.0^2} = e^{1} \\approx 2.718282 \\\\ 3 & 1.5 & e^{1.5^2} = e^{2.25} \\approx 9.487736 \\\\ 4 & 2.0 & e^{2.0^2} = e^4 \\approx 54.598150 \\\\ \\hline \\end{array}$$"},{"type":"text","order":3,"title":"Apply trapezium rule","content":"The trapezium rule formula for 4 strips over $[0, 2]$ is:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx \\frac{h}{2} \\left( y_0 + y_4 + 2(y_1 + y_2 + y_3) \\right)$$\\n\\nSubstituting our values into the formula:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx \\frac{0.5}{2} \\left( 1 + e^4 + 2(e^{0.25} + e + e^{2.25}) \\right)$$\\n\\nUsing a calculator to compute the sum inside the brackets:\\n\\n$$\\begin{aligned} \\text{Sum} &= 1 + 54.598150 + 2(1.284025 + 2.718282 + 9.487736) \\\\ &= 55.598150 + 2(13.490043) \\\\ &= 55.598150 + 26.980086 \\\\ &= 82.578236 \\end{aligned}$$\\n\\nNow, multiply by $\\frac{0.5}{2} = 0.25$:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx 0.25 \\times 82.578236 = 20.644559$$","calculator":[{"gdc":{"expression":"0.25 * (e^(0) + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))","instructions":"In the Run-Matrix menu, type the calculation directly:\n0.25 * (1 + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))"},"ti84":{"expression":"0.25 * (e^(0) + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))","instructions":"Enter the expression on the home screen:\n0.25 * (1 + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))"},"desmos":{"expression":"0.25 * (1 + e^4 + 2 * (e^0.25 + e^1 + e^2.25))","instructions":"Type the following expression to get the result:\n0.25 * (1 + e^4 + 2 * (e^0.25 + e^1 + e^2.25))"},"description":"Evaluate the one-sided trapezium rule sum."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int_{-2}^2 e^{x^2} \\, dx = \\htmlData{anchor=double-factor}{\\color{@sky}{2}} \\int_{0}^2 e^{x^2} \\, dx$"},{"latex":"$$\\phantom{\\int_{-2}^2 e^{x^2} \\, dx} \\approx \\htmlData{anchor=double-apply}{\\color{@sky}{2}} \\times \\htmlData{anchor=half-val}{\\color{@amber}{20.644559}}$"},{"latex":"$$\\phantom{\\int_{-2}^2 e^{x^2} \\, dx} = \\htmlData{anchor=final-val}{\\color{@purple}{41.289118}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"double-factor"},{"color":"amber","style":"text-color","anchor":"half-val"},{"color":"purple","style":"text-color","anchor":"final-val"}],"connections":[{"to":"double-apply","from":"double-factor","color":"sky","label":"double","style":"solid"}]},"order":4,"title":"Double the result","description":"Now, we double our estimate from the interval $[0, 2]$ to find the total approximate value of the integral from $-2$ to $2$."},{"type":"text","order":5,"title":"State final answer","content":"Rounding our result to 3 significant figures:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx \\approx 41.3 \\text{ (3 s.f.)}$$\\n\\nThis matches the standard application of the trapezium rule across all 8 strips, but utilizing symmetry halved the number of function evaluations needed!"}]}],"generatedAt":"2026-06-09T03:52:03.700Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"1934f05f-e699-4096-b6ae-0b1793258fd1:direct_trapezium","label":"Direct Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the step size","content":"The trapezium rule is used to estimate the area under a curve by dividing it into a series of strips. First, we need to calculate the width of each strip, known as the step size $h$.\\n\\nThe formula for the step size is:\\n\\n$$h = \\frac{b - a}{n}$$\\n\\nwhere:\\n- $a = -2$ (the lower limit of integration)\\n- $b = 2$ (the upper limit of integration)\\n- $n = 8$ (the number of strips)\\n\\nSubstituting these values in gives:\\n\\n$$h = \\frac{2 - (-2)}{8} = \\frac{4}{8} = 0.5$$"},{"type":"text","order":1,"title":"Identify the x-values","content":"Using our step size of $h = 0.5$, we list all the $x$-coordinates starting from the lower limit $x_0 = -2$ up to the upper limit $x_8 = 2$:\\n\\n$$\\begin{aligned} x_0 &= -2 \\\\ x_1 &= -1.5 \\\\ x_2 &= -1 \\\\ x_3 &= -0.5 \\\\ x_4 &= 0 \\\\ x_5 &= 0.5 \\\\ x_6 &= 1 \\\\ x_7 &= 1.5 \\\\ x_8 &= 2 \\end{aligned}$$"},{"type":"text","order":2,"title":"Evaluate the function","content":"Now, we evaluate our function $f(x) = e^{x^2}$ at each of these $x$-values. It is best to keep these in exact exponential form to avoid rounding errors during the intermediate steps:\\n\\n$$\\begin{aligned} y_0 &= f(-2) = e^{(-2)^2} = e^4 \\\\ y_1 &= f(-1.5) = e^{(-1.5)^2} = e^{2.25} \\\\ y_2 &= f(-1) = e^{(-1)^2} = e^1 \\\\ y_3 &= f(-0.5) = e^{(-0.5)^2} = e^{0.25} \\\\ y_4 &= f(0) = e^{0^2} = e^0 = 1 \\\\ y_5 &= f(0.5) = e^{0.5^2} = e^{0.25} \\\\ y_6 &= f(1) = e^{1^2} = e^1 \\\\ y_7 &= f(1.5) = e^{1.5^2} = e^{2.25} \\\\ y_8 &= f(2) = e^{2^2} = e^4 \\end{aligned}$$"},{"type":"text","order":3,"title":"Apply trapezium rule formula","content":"The general formula for the trapezium rule is:\\n\\n$$\\int_{a}^{b} f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\\n\\nAlternatively, we can write this directly as a sum of individual trapeziums:\\n\\n$$\\int_{-2}^{2} e^{x^2} \\, dx \\approx h \\left( \\frac{y_0}{2} + y_1 + y_2 + y_3 + y_4 + y_5 + y_6 + y_7 + \\frac{y_8}{2} \\right)$$\\n\\nSubstituting our values into this formula:\\n\\n$$\\text{Estimate} = 0.5 \\left( \\frac{e^4}{2} + e^{2.25} + e^1 + e^{0.25} + 1 + e^{0.25} + e^1 + e^{2.25} + \\frac{e^4}{2} \\right)$$"},{"type":"text","order":4,"title":"Calculate the final estimate","content":"Combining the matching terms simplifies our calculation:\\n\\n$$\\text{Estimate} = 0.5 \\left[ e^4 + 2e^{2.25} + 2e + 2e^{0.25} + 1 \\right]$$\\n\\nNow, we calculate the numerical values using a scientific calculator:\\n- $e^4 \\approx 54.5982$\\n- $2e^{2.25} \\approx 2 \\times 9.4877 = 18.9755$\\n- $2e^1 \\approx 2 \\times 2.7183 = 5.4366$\\n- $2e^{0.25} \\approx 2 \\times 1.2840 = 2.5681$\\n- $1$\\n\\nSumming these values:\\n\\n$$54.5982 + 18.9755 + 5.4366 + 2.5681 + 1 = 82.5784$$\\n\\nFinally, multiply by $0.5$:\\n\\n$$\\text{Estimate} = 0.5 \\times 82.5784 = 41.2892$$\\n\\nRounding to $3$ significant figures, we get:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx \\approx 41.3$$","calculator":[{"gdc":{"expression":"0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)","instructions":"Enter the expression into the run-matrix menu:\n0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)"},"ti84":{"expression":"0.5 * (e^(4) + 2*e^(2.25) + 2*e^(1) + 2*e^(0.25) + 1)","instructions":"Type the simplified sum directly on the home screen:\nPress [0.5] [*(] [2nd] [LN] (e^x) [4] [)] [+] [2] [2nd] [LN] [2.25] [)] [+] [2] [2nd] [LN] [1] [)] [+] [2] [2nd] [LN] [0.25] [)] [+] [1] [Enter]."},"desmos":{"expression":"0.5 * (e^{4} + 2e^{2.25} + 2e^{1} + 2e^{0.25} + 1)","instructions":"Type the direct expression into a cell to get the calculated decimal value."},"description":"Evaluate the definite integral approximation directly on a graphing calculator or tool."}]}]},{"id":"1934f05f-e699-4096-b6ae-0b1793258fd1:symmetry_trapezium","label":"Using Symmetry","steps":[{"type":"text","order":0,"title":"Exploit function symmetry","content":"The integrand function $f(x) = e^{x^2}$ is an **even function**, because:\\n\\n$$f(-x) = e^{(-x)^2} = e^{x^2} = f(x)$$\\n\\nDue to this symmetry about the $y$-axis, the area under the curve from $-2$ to $2$ is exactly double the area from $0$ to $2$:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx = 2 \\int_{0}^2 e^{x^2} \\, dx$$\\n\\nInstead of applying the trapezium rule with 8 strips over $[-2, 2]$, we can apply it with 4 strips over $[0, 2]$ and then multiply our final estimate by 2. This significantly reduces our calculation steps and minimizes errors."},{"type":"text","order":1,"title":"Calculate step size","content":"For 4 strips over the interval $[0, 2]$, we calculate the step size $h$ using the formula:\\n\\n$$h = \\frac{b - a}{n}$$\\n\\nSubstituting our values:\\n\\n$$h = \\frac{2 - 0}{4} = 0.5$$"},{"type":"text","order":2,"title":"Create coordinate table","content":"Starting at $x_0 = 0$ and adding $h = 0.5$ at each step, we find the coordinates and evaluate $y = e^{x^2}$:\\n\\n$$\\begin{array}{|c|l|l|} \\hline i & x_i & y_i = e^{x_i^2} \\\\ \\hline 0 & 0 & e^{0^2} = 1 \\\\ 1 & 0.5 & e^{0.5^2} = e^{0.25} \\approx 1.284025 \\\\ 2 & 1.0 & e^{1.0^2} = e^{1} \\approx 2.718282 \\\\ 3 & 1.5 & e^{1.5^2} = e^{2.25} \\approx 9.487736 \\\\ 4 & 2.0 & e^{2.0^2} = e^4 \\approx 54.598150 \\\\ \\hline \\end{array}$$"},{"type":"text","order":3,"title":"Apply trapezium rule","content":"The trapezium rule formula for 4 strips over $[0, 2]$ is:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx \\frac{h}{2} \\left( y_0 + y_4 + 2(y_1 + y_2 + y_3) \\right)$$\\n\\nSubstituting our values into the formula:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx \\frac{0.5}{2} \\left( 1 + e^4 + 2(e^{0.25} + e + e^{2.25}) \\right)$$\\n\\nUsing a calculator to compute the sum inside the brackets:\\n\\n$$\\begin{aligned} \\text{Sum} &= 1 + 54.598150 + 2(1.284025 + 2.718282 + 9.487736) \\\\ &= 55.598150 + 2(13.490043) \\\\ &= 55.598150 + 26.980086 \\\\ &= 82.578236 \\end{aligned}$$\\n\\nNow, multiply by $\\frac{0.5}{2} = 0.25$:\\n\\n$$\\int_{0}^2 e^{x^2} \\, dx \\approx 0.25 \\times 82.578236 = 20.644559$$","calculator":[{"gdc":{"expression":"0.25 * (e^(0) + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))","instructions":"In the Run-Matrix menu, type the calculation directly:\n0.25 * (1 + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))"},"ti84":{"expression":"0.25 * (e^(0) + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))","instructions":"Enter the expression on the home screen:\n0.25 * (1 + e^(4) + 2 * (e^(0.25) + e^(1) + e^(2.25)))"},"desmos":{"expression":"0.25 * (1 + e^4 + 2 * (e^0.25 + e^1 + e^2.25))","instructions":"Type the following expression to get the result:\n0.25 * (1 + e^4 + 2 * (e^0.25 + e^1 + e^2.25))"},"description":"Evaluate the one-sided trapezium rule sum."}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\int_{-2}^2 e^{x^2} \\, dx = \\htmlData{anchor=double-factor}{\\color{@sky}{2}} \\int_{0}^2 e^{x^2} \\, dx$"},{"latex":"$$\\phantom{\\int_{-2}^2 e^{x^2} \\, dx} \\approx \\htmlData{anchor=double-apply}{\\color{@sky}{2}} \\times \\htmlData{anchor=half-val}{\\color{@amber}{20.644559}}$"},{"latex":"$$\\phantom{\\int_{-2}^2 e^{x^2} \\, dx} = \\htmlData{anchor=final-val}{\\color{@purple}{41.289118}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"double-factor"},{"color":"amber","style":"text-color","anchor":"half-val"},{"color":"purple","style":"text-color","anchor":"final-val"}],"connections":[{"to":"double-apply","from":"double-factor","color":"sky","label":"double","style":"solid"}]},"order":4,"title":"Double the result","description":"Now, we double our estimate from the interval $[0, 2]$ to find the total approximate value of the integral from $-2$ to $2$."},{"type":"text","order":5,"title":"State final answer","content":"Rounding our result to 3 significant figures:\\n\\n$$\\int_{-2}^2 e^{x^2} \\, dx \\approx 41.3 \\text{ (3 s.f.)}$$\\n\\nThis matches the standard application of the trapezium rule across all 8 strips, but utilizing symmetry halved the number of function evaluations needed!"}]}],"generatedAt":"2026-06-09T03:52:41.963Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"38571646-66bb-49bf-8b1a-8ae3585977e4","revisionId":"af9b3cbd-a238-4b12-8cf7-ca029cb29b12","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"4ec99a15-343e-41ef-babe-cbac66b4ffa1","label":"a","content":"Use the trapezium rule with 6 strips to estimate the value of\n\n$$\\int_1^{4} x \\ln(x + 1) \\, dx$$","metadata":null,"markscheme":"- Calculate step size $h = \\frac{4 - 1}{6} = 0.5$ M1\n- Evaluate function at $x = 1, 1.5, 2, 2.5, 3, 3.5, 4$ M1\n- Values: $0.6931, 1.3744, 2.1972, 3.1319, 4.1589, 5.2643, 6.4378$ M1\n- Apply formula: $\\frac{0.5}{2} [0.6931 + 6.4378 + 2(1.3744 + 2.1972 + 3.1319 + 4.1589 + 5.2643)]$ M2\n- Final answer $9.85$ (3sf) A1","marks":6,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Calculation of step size $h = 0.5$.","code":"M"},{"id":"p2","marks":1,"description":"Correct $x$ values identified ($1, 1.5, 2, 2.5, 3, 3.5, 4$).","code":"M"},{"id":"p3","marks":1,"description":"Function values $y_i$ calculated correctly to at least 3dp.","code":"M"},{"id":"p4","marks":2,"description":"Correct application of the trapezium rule formula.","code":"M"},{"id":"p5","marks":1,"description":"Final answer of 9.85 (accept 9.846).","code":"A"}],"finalAnswers":["9.85"],"generatedAt":"2026-05-06T14:04:50.038Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"trapezium_rule","label":"Standard Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the strip width","content":"To begin, we find the width of each strip, $h$, using the formula:\n\n$$h = \\frac{b - a}{n}$$\n\nwhere $a$ and $b$ are the lower and upper limits of integration, and $n$ is the number of strips.\n\nHere, our limits are $a = 1$ and $b = 4$, and we are given $n = 6$ strips:\n\n$$h = \\frac{4 - 1}{6} = \\frac{3}{6} = 0.5$$","highlights":[{"text":"6 strips","location":"specification"}]},{"type":"text","order":1,"title":"Construct the coordinates table","content":"Next, we generate the $x$-coordinates starting at the lower limit $x_0 = 1$ and increasing by our step size $h = 0.5$ up to the upper limit $x_6 = 4$. \n\nWe then evaluate the function $f(x) = x \\ln(x + 1)$ at each of these $x$-values (keeping at least $4$ decimal places for accuracy):\n\n| $i$ | $x_i$ | $y_i = x_i \\ln(x_i + 1)$ |\n|---|---|---|\n| $0$ | $1$ | $1 \\ln(2) \\approx 0.6931$ |\n| $1$ | $1.5$ | $1.5 \\ln(2.5) \\approx 1.3744$ |\n| $2$ | $2$ | $2 \\ln(3) \\approx 2.1972$ |\n| $3$ | $2.5$ | $2.5 \\ln(3.5) \\approx 3.1319$ |\n| $4$ | $3$ | $3 \\ln(4) \\approx 4.1589$ |\n| $5$ | $3.5$ | $3.5 \\ln(4.5) \\approx 5.2642$ |\n| $6$ | $4$ | $4 \\ln(5) \\approx 6.4378$ |"},{"type":"text","order":2,"title":"Apply the trapezium rule","content":"The formula for the trapezium rule is:\n\n$$\\int_a^b f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nSubstituting our calculated values into the formula:\n\n$$\\text{Estimate} = \\frac{0.5}{2} \\left[ 0.6931 + 6.4378 + 2(1.3744 + 2.1972 + 3.1319 + 4.1589 + 5.2642) \\right]$$"},{"type":"text","order":3,"title":"Calculate the final estimate","content":"First, we sum the middle terms:\n\n$$1.3744 + 2.1972 + 3.1319 + 4.1589 + 5.2642 = 16.1266$$\n\nNow, substitute this sum back into the expression:\n\n$$\\text{Estimate} = 0.25 \\times [7.1309 + 2(16.1266)]$$\n\n$$\\text{Estimate} = 0.25 \\times [7.1309 + 32.2532]$$\n\n$$\\text{Estimate} = 0.25 \\times 39.3841$$\n\n$$\\text{Estimate} \\approx 9.8460$$\n\nRounding to $3$ significant figures gives:\n\n$$\\text{Estimate} \\approx 9.85$$"}]}],"generatedAt":"2026-06-09T02:32:26.545Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4ec99a15-343e-41ef-babe-cbac66b4ffa1:trapezium_rule","label":"Standard Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the strip width","content":"To begin, we find the width of each strip, $h$, using the formula:\n\n$$h = \\frac{b - a}{n}$$\n\nwhere $a$ and $b$ are the lower and upper limits of integration, and $n$ is the number of strips.\n\nHere, our limits are $a = 1$ and $b = 4$, and we are given $n = 6$ strips:\n\n$$h = \\frac{4 - 1}{6} = \\frac{3}{6} = 0.5$$","highlights":[{"text":"6 strips","location":"specification"}]},{"type":"text","order":1,"title":"Construct the coordinates table","content":"Next, we generate the $x$-coordinates starting at the lower limit $x_0 = 1$ and increasing by our step size $h = 0.5$ up to the upper limit $x_6 = 4$. \n\nWe then evaluate the function $f(x) = x \\ln(x + 1)$ at each of these $x$-values (keeping at least $4$ decimal places for accuracy):\n\n| $i$ | $x_i$ | $y_i = x_i \\ln(x_i + 1)$ |\n|---|---|---|\n| $0$ | $1$ | $1 \\ln(2) \\approx 0.6931$ |\n| $1$ | $1.5$ | $1.5 \\ln(2.5) \\approx 1.3744$ |\n| $2$ | $2$ | $2 \\ln(3) \\approx 2.1972$ |\n| $3$ | $2.5$ | $2.5 \\ln(3.5) \\approx 3.1319$ |\n| $4$ | $3$ | $3 \\ln(4) \\approx 4.1589$ |\n| $5$ | $3.5$ | $3.5 \\ln(4.5) \\approx 5.2643$ |\n| $6$ | $4$ | $4 \\ln(5) \\approx 6.4378$ |"},{"type":"text","order":2,"title":"Apply the trapezium rule","content":"The formula for the trapezium rule is:\n\n$$\\int_a^b f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nSubstituting our calculated values into the formula:\n\n$$\\text{Estimate} = \\frac{0.5}{2} \\left[ 0.6931 + 6.4378 + 2(1.3744 + 2.1972 + 3.1319 + 4.1589 + 5.2643) \\right]$$"},{"type":"text","order":3,"title":"Calculate the final estimate","content":"First, we sum the middle terms:\n\n$$1.3744 + 2.1972 + 3.1319 + 4.1589 + 5.2643 = 16.1267$$\n\nNow, substitute this sum back into the expression:\n\n$$\\text{Estimate} = 0.25 \\times [7.1309 + 2(16.1267)]$$\n\n$$\\text{Estimate} = 0.25 \\times [7.1309 + 32.2534]$$\n\n$$\\text{Estimate} = 0.25 \\times 39.3843$$\n\n$$\\text{Estimate} \\approx 9.8461$$\n\nRounding to $3$ significant figures gives:\n\n$$\\text{Estimate} \\approx 9.85$$"}]}],"generatedAt":"2026-06-09T02:32:35.247Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths","summaries":[{"id":"a3f4543a-299a-4a9e-91ff-cd3ea231da7f","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"007e90cd-b1b6-4fe4-9394-d4c91332ccab","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2380f38c-c2db-4e0d-8d0f-00e6989b11fa","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"e2c2c03e-bea9-4918-88fe-109d91d95b2b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f87a257b-ab55-4968-8341-19688d307a9e","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0c24a866-78aa-4003-a555-7ff1ec326c79","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"397dbad6-42d3-433c-acb9-f68b49d4ff96","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"3e4af7a7-b99a-4f21-ab56-e20594ea0496","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"3ed1a603-2601-4f22-9eaf-2e96a3c1d462","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"addf3047-e2b1-4aff-a52b-dae0e09d5006","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"2469c899-7969-4983-932e-a0d7e5a414ee","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"38571646-66bb-49bf-8b1a-8ae3585977e4","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"85b5bb78-076f-4ba6-949d-df19181f589c","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f816cef1-764b-45b4-b72c-7afe73af6948","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"f9f073bb-73b6-48cd-ab6d-44f22f529058","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"bd7f0ba9-d9ff-4d58-9b94-766b3267e563","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"09296e6e-94eb-4835-95d7-58986d1f0476","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"118ea241-7046-409e-b97b-584917cb86a0","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"18470c9f-0e26-445d-8fba-20e85521e062","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"1c094bfc-81a0-499d-816d-849be8d9f8d4","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"244102ec-b9cd-4011-8acb-e970dfc01abf","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"28a4af6d-dacd-4a8d-bf79-74735466d884","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"356c52a9-7f43-4202-a261-08ae94f943b7","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"45b79365-fea8-4159-a172-d466ab58556d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"47e5cdb2-ee53-4aba-93fe-9f1ec53553ee","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"57e3cee4-5fa7-4131-bae7-04906b5eb67d","difficultyBand":"hard","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"5eefaac2-0bfd-4caa-ad85-c730670bb99f","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"6617d362-2c7c-4efa-898a-b0bc82cc179c","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7c823f12-bb5e-43d4-b10f-529e30c14cbd","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"85da4b6c-56d1-464f-9790-bfb27ef26ebe","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"98033fb8-e5a2-478d-a197-f8b30993bbb6","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"b3ffcedc-9e39-411c-a5a1-3dfb8c20eb55","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b4e64880-fb5c-4b29-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The Trapezium Rule","topicId":"1575b9fa-8d09-465b-8f78-79c2e5c1591b","topicSidebar":"$L92","topicSlug":"a-level-pure-mathematics-11-9-the-trapezium-rule","topicTitle":"11.9 The Trapezium Rule"}]}],"$L93","$L94"]}]

11.9 The Trapezium Rule

Question 10

11.9 The Trapezium Rule Questions

11.9 The Trapezium Rule

Question 10

11.9 The Trapezium Rule Questions