grals and areas Flashcards","topicId":"caf30ef5-73cc-43a7-9293-95b0129f84d2","topicSlug":"definite-integrals-and-areas","cards":[{"id":"d0972678-694f-40de-a52a-51fa8608b5b0","cardType":"basic","front":"What does $dx$ indicate in $\\int_a^b f(x)\\,dx$?","back":"Integrate with respect to $x$.","points":1,"mcqOptions":null},{"id":"e2a2e444-0a80-4918-840c-5c3a7b31fccb","cardType":"cloze","front":"In $\\int_a^b f(x)\\,dx$, $a$ is the [ ].","back":"In $\\int_a^b f(x)\\,dx$, $a$ is the lower limit.","points":null,"mcqOptions":null},{"id":"7af90415-3b52-4d7a-a92d-4e49ec6792bc","cardType":"cloze","front":"In $\\int_a^b f(x)\\,dx$, $b$ is the [ ].","back":"In $\\int_a^b f(x)\\,dx$, $b$ is the upper limit.","points":null,"mcqOptions":null},{"id":"6c970188-1463-4b1a-a53c-d9bd6076916e","cardType":"cloze","front":"If $F(x)$ is an antiderivative of $f(x)$, then $\\text{[...]}$.","back":"If $F(x)$ is an antiderivative of $f(x)$, then $\\boxed{F'(x)=f(x)}$.","points":null,"mcqOptions":null},{"id":"fe3dda0b-496e-4067-9129-f3502a1b67fd","cardType":"cloze","front":"The Fundamental Theorem of Calculus gives $\\text{[...]}$.","back":"The Fundamental Theorem of Calculus gives $\\boxed{\\displaystyle \\int_a^b f(x)\\,dx=F(b)-F(a)}$.","points":null,"mcqOptions":null},{"id":"2a5d6a1a-5606-43c2-91c1-d1f2188e8ee1","cardType":"basic","front":"In what order are limits substituted into an antiderivative?","back":"Upper minus lower: $F(b)-F(a)$.","points":1,"mcqOptions":null},{"id":"7a363804-99c7-4bc4-982c-9b961cc704a6","cardType":"basic","front":"What does $[F(x)]_a^b$ mean?","back":"$F(b)-F(a)$.","points":1,"mcqOptions":null},{"id":"d1aa0670-5b81-4918-9aba-b579c7796e55","cardType":"basic","front":"Why is $+c$ omitted from a definite integral?","back":"The constant appears in both terms and cancels.","points":2,"mcqOptions":null},{"id":"cc540151-753c-42ee-978a-8401539fb249","cardType":"basic","front":"Evaluate $\\displaystyle \\int_1^3(2x^2-4x+5)\\,dx$.","back":"$\\dfrac{34}{3}$.","points":2,"mcqOptions":null},{"id":"0ab38e9f-c5f6-45a1-b387-a4b62862c100","cardType":"cloze","front":"A region above the $x$-axis contributes a [ ] signed area.","back":"A region above the $x$-axis contributes a positive signed area.","points":null,"mcqOptions":null},{"id":"2df87b39-fbce-4c61-89e3-aa8cc5145b65","cardType":"cloze","front":"A region below the $x$-axis contributes a [ ] signed area.","back":"A region below the $x$-axis contributes a negative signed area.","points":null,"mcqOptions":null},{"id":"c3f75655-2e53-4749-bc9c-f1e790168f5f","cardType":"basic","front":"Why can a definite integral differ from total geometric area?","back":"Positive and negative signed areas can cancel.","points":2,"mcqOptions":null},{"id":"4bbd7b8e-01b3-4c2a-b798-cddea2969953","cardType":"basic","front":"If $\\displaystyle \\int_0^2(x-3)\\,dx=-4$, what is the geometric area?","back":"$4$ square units.","points":1,"mcqOptions":null},{"id":"eb907d2a-ded7-4292-a8fc-131d2ab16247","cardType":"basic","front":"If $f(x)\\geq0$ on $[a,b]$, how is the area found?","back":"$\\displaystyle A=\\int_a^b f(x)\\,dx$.","points":1,"mcqOptions":null},{"id":"cae61e65-7a47-450a-807d-1dbedb92612c","cardType":"basic","front":"If $f(x)\\leq0$ on $[a,b]$, how is the geometric area found?","back":"$\\displaystyle A=-\\int_a^b f(x)\\,dx$.","points":1,"mcqOptions":null},{"id":"a7e72a45-e301-43d5-a2ec-5a452a60296f","cardType":"basic","front":"Where must an integral be split when finding total area against the $x$-axis?","back":"At eve

1.11.3 Definite integrals and areas Flashcards

1.11.3 Definite integrals and areas Flashcards