gration (A-level only) Flashcards","topicId":"cf0327d4-5636-4f73-93c9-2ec1bda7cff1","topicSlug":"numerical-integration","cards":[{"id":"e8e4c65f-b017-4b21-8e10-bddd5e65ec6b","cardType":"basic","front":"

What method estimates a definite integral when an exact antiderivative is unavailable?

","back":"Use numerical integration.","points":1,"mcqOptions":null},{"id":"4965edbe-464f-4223-a854-921203101a70","cardType":"basic","front":"

How does a definite integral treat a region below the $x$-axis?

","back":"It counts the region as a negative signed area.","points":1,"mcqOptions":null},{"id":"52353fbc-ef6f-4b14-b330-44153d28b860","cardType":"basic","front":"

How should total geometric area be found when a curve crosses the $x$-axis?

","back":"Split at the roots and add each region's positive area.","points":2,"mcqOptions":null},{"id":"5e8d24a8-fed0-47e1-b6ad-756552fa6d7a","cardType":"cloze","front":"

For $n$ equal strips from $a$ to $b$, the strip width is $\\text{[...]}$.

","back":"

For $n$ equal strips from $a$ to $b$, the strip width is $\\boxed{h=\\frac{b-a}{n}}$.

","points":null,"mcqOptions":null},{"id":"0ca608d0-2051-4ebd-a81c-0ff6fa3a8e36","cardType":"basic","front":"

Seven ordinates are used from $x=1$ to $x=4$. What is the strip width?

","back":"There are six strips, so $h=\\frac{4-1}{6}=0.5$.","points":2,"mcqOptions":null},{"id":"7d76c7d1-befa-4033-89ce-d2d8359daf3f","cardType":"basic","front":"

How many ordinates are needed for $n$ strips?

","back":"$n+1$ ordinates.","points":1,"mcqOptions":null},{"id":"c908a6bb-05f8-409e-bebd-1c384aa6614b","cardType":"cloze","front":"

The ordinate at $x=x_r$ is $\\text{[...]}$.

","back":"

The ordinate at $x=x_r$ is $\\boxed{y_r=f(x_r)}$.

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What is the area used for the first trapezium of width $h$?

","back":"$\\frac{1}{2}h(y_0+y_1)$","points":1,"mcqOptions":null},{"id":"7f19a67f-2e1b-44db-af1a-eb6fe231a1da","cardType":"cloze","front":"

The trapezium rule is $$\\text{[...]}$$.

","back":"

The trapezium rule is $$\\boxed{\\int_a^b f(x)\\,dx\\approx\\frac{h}{2}\\left[y_0+y_n+2(y_1+\\cdots+y_{n-1})\\right]}$$.

","points":null,"mcqOptions":null},{"id":"e3080434-39c3-4b9b-840c-0000408fb37c","cardType":"basic","front":"

Which ordinates have coefficient 2 in the trapezium rule?

","back":"All middle ordinates, from $y_1$ to $y_{n-1}$.","points":1,"mcqOptions":null},{"id":"fd69b74d-ea6d-4edd-930c-9f3568502187","cardType":"basic","front":"

Why should calculated ordinates not be rounded early?

","back":"Early rounding can noticeably change the final estimate.","points":2,"mcqOptions":null},{"id":"09e49037-332e-43ac-acf0-db92ecebd73a","cardType":"basic","front":"

Why does using more equal strips usually improve a trapezium-rule estimate?

","back":"The narrower chords follow the curve more closely.","points":2,"mcqOptions":null},{"id":"1fb5630c-7756-40b1-a2bf-516d957303fd","cardType":"cloze","front":"

If $f''(x)>0$ throughout an interval, the curve is [ ].

","back":"

If $f''(x)>0$ throughout an interval, the curve is concave up.

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Why does the trapezium rule overestimate for a concave-up curve?

","back":"The straight chords lie above the curve.","points":2,"mcqOptions":null},{"id":"a421dd27-7f6d-4b3d-a650-7fd6b6726eb3","cardType":"basic","front":"

Why does the trapezium rule underestimate for a concave-down curve?

","back":"The straight chords lie below the curve.","points":2,"mcqOptions":null},{"id":"d42de087-180f-4e3e-a113-29b5840cfc28","cardType":"basic","front":"

For $f(x)=x^3+2x$ on $1\\leq

1.9.6 Numerical integration (A-level only) Flashcards

1.9.6 Numerical integration (A-level only) Flashcards