AP Calculus BC Chapter 5 Practice Test - Definite and Indefinite Integrals

Sep 30, 2026

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AP Calculus BC Chapter 5 practice test focused on approximating areas under curves with left-endpoint rectangles and correcting estimates using exact integration. It includes right-hand Riemann sum approximations from tabulated function values, evaluation of multiple indefinite integrals (including trigonometric and substitution forms), and several definite integrals with piecewise definitions. A motion application uses a position function to compute displacement and distance traveled, followed by a full answer key.

LanguageEnglish

CategoryExam

FormatPDF

Pages5

Reading Time13 min

AccessFree to download

Chapter 5 Practice Test

Area under the curve with 4 left rectangles and exact integration

Riemann sum approximation using right endpoints

Evaluate indefinite integrals

Evaluate definite integrals

Find F'(x) from given F(x)

Particle motion: displacement and distance traveled

Answers to Chapter 5 Practice Test

ID: 1 AP Calculus BC Name___________________________________ Chapter 5 Practice Test Date________________ Period____ © 201 3 Ksuta S ofwtwa re LLMC . APll rights re servied . For each problem, approximate the area under the curve over the given interval using 4 left endpoint rectangles. For comparison, also find the exact area using integration. Explain why your answer is an over-or under-estimate. 1) y = −x + 3; [0, 2] 2 2) y = − ; [−4, −2] x For each problem, use a right-hand Riemann sum to approximate the integral based off of the values in the table. 3) ∫019 f (x) dx | x | 0 | 2 | 4 | 5 | 6 | 7 | 19 | | --- | --- | --- | --- | --- | --- | --- | --- | | f ( x) | 3 | 2 | 1 | 3 | 1 | 2 | 1 | 4) ∫020 f (x) dx | x | 0 | 1 | 3 | 13 | 15 | 18 | 20 | | --- | --- | --- | --- | --- | --- | --- | --- | | f ( x) | 7 | 8 | 7 | 5 | 3 | 2 | 0 | Evaluate each indefinite integral. 5) ∫ ~~ ~~15x4 dx 6) ∫ − ~~ ~~2544~~ ~~x dx 7) ∫ (−5x32 − 7x34~~ ~~ ) dx 8) ∫ ~~ ~~2(~~ ~~12x8~~ ~~83x~~ ~~6~~ ~~+~~ ~~1)~~ ~~ dx © 2 01 3 KDu t a S o fPt war e LLWC . AlMlc riEght sR reserved. Mad e w irt hX InffdiJni5t e C ajl1culyu s . -1- 9) ∫ ~~ ~~csinos2xx dx 10) ∫ ~~ ~~co10s2~~ ~~x dx 11) ∫ ~~ ~~7scoins2~~ ~~xx dx 12) ∫ ~~ ~~se1c~~ ~~x dx 13) ∫ 6sec 3x ⋅ tan 3xsec 3x dx 14) ∫ (4x3 + 1)4 ⋅ 36x2 dx 15) ∫ ~~ ~~(3x121)~~ ~~5~~ ~~ dx 16) ∫ ~~ ~~6scoins3~~ ~~3x3x dx 17) ∫ ~~ ~~sin~~ ~~2~~ ~~6(0x3x45~~ ~~+~~ ~~2) dx 18) ∫ ~~ ~~csc~~ ~~1(255xx5~~ ~~3)~~ ~~ dx 19) ∫ ~~ ~~20csc~~ ~~−5xco⋅~~ ~~cso2t(−5cscxsi5xn)~~ ~~(csc~~ ~~−5x) dx 20) ∫ ~~ ~~s4in⋅2se(tc2anxx) dx © 2 0 13 KLu t a Softt wja r e LLC . AlLlK rijgh t sW reser ved.W MaEdje wikt h Intf9i3nGiut e C al2cAul4us . -2- Evaluate each definite integral. 21) −31 (−x + −3x − 6 ) dx 23) 13 x 2 + x dx 25) 14f (x) dx, f (x) = {xx 33,, xx −22 27) 32−(x + 3)~~ ~~13 dx 29) ∫01~~ ~~(4x223)2~~ ~~ dx © 2 0 1 3 Kujta S ofNt w ar e LjL Cp. ADlYlq riHg hxt sG re serv ed . MAaMd e wi0t h I nfzi9ni3te CaMlpcu luu s. π − 22) ∫ 2 csc x ⋅ cot x dx 2π − 3 24) 27f ( x) dx, f (x) = {x32, + 10x + 24, xx −33 26) 21(x4 − x2 − x − 4) dx 28) 33f (x) dx, f (x) = {−2x2x− ,1, 30) ∫01−18x2 (3x3 − 1)2 dx x ≤ 2 x > 2 -3- 31) ∫02 − ~~ ~~(x2~~ ~~2)~~ ~~2~~ ~~ dx 32) ∫014x(x2 − 1)2 dx For each problem, find F'(x). 33) F (x) = x14t dt 34) F (x) = 25x4(t + 3)~~ ~~12 dt 2 2 35) F (x) = ∫xx t~~ ~~12 dt 36) F (x) = ∫2x − ~~ ~~t13~~ ~~ dt 37) F (x) = ∫0x3 − ~~ ~~(t~~ ~~+42)2~~ ~~ dt 38) F (x) = ∫x2x(−t2 − 6t − 3) dt A particle moves along a horizontal line. Its position function, in feet, is s(t) for t ≥ 0 seconds. For each problem, find the displacement of the particle and the distance traveled by the particle over the given interval. 39) s(t) = t3 − 28t2 + 196t; 1 ≤ t ≤ 9 40) s(t) = −t3 + 15t2 ; 9 ≤ t ≤ 15 © 2 0 1 3 Ku ta Soflt war e LLACH. AflJl0 ridgh t s reserved . Made wipthT I n fBiyn iYte Caldculgu sC. -4- Answers to Chapter 5 Practice Test (ID: 1) 1) Exact: 4 2) Exact: −2 ln 2 + 2 ln 4 ≈ 1.386 3) 24 Approximate: ~~ ~~92 = 4.5 Approximate: ~~ ~~533420 ≈ 1.269 4) 84 5 5 5 7 5) − x 3 + C 6) −5x 4 + C 7) −2x 2 − 4x 4 + C 6 4 1 9) sec x + C 10) 10tan x + C 11) −7csc x + C 8) 4x + 4x − ~~ ~~ + Cx2 12) sin x + C 13) 32 ⋅ (sec 3x)43 + C 14) 35(4x3 + 1) 5 + C 15) − ~~ ~~2(3x21+~~ ~~1)4~~ ~~ + C 1 17) −4cot (3x5 + 2) + C 18) −5cos (5x5 − 3) + C 16) − ~~ ~~ + C sin 2 3x 19) 4sec (csc −5x) + C 20) cot (tan −4x) + C 21) 7 3 − 2 3 22) 3 ≈ −0.155 23) 173 ≈ 5.667 24) 493 ≈ 16.333 25) − ~~ ~~272 = −13.5 26) − ~~ ~~9910 = −9.9 3 27) −18~~ ~~4~~ ~~+~~ ~~3 ≈ −7.427 28) − ~~ ~~194 = −4.75 29) 47 ≈ 0.571 30) −6 2 2 1 31) − 3 ≈ −0.667 32) 3 ≈ 0.667 33) F ' (x) = x 1 1 2 12x2 34) F ' (x) = 8(2x + 3) 2 35) F ' (x) = 2x2 − x 2 36) F ' (x) = − x 5 37) F ' (x) = − ( x3 + 2) 2 38) F ' (x) = −7x2 − 18x − 3 39) Displacement: 56 Distance traveled: ~~ ~~1131274 ≈ 419.037 40) Displacement: −486 Distan

FAQs

How are areas under the curve approximated in this test?

Use 4 left endpoint rectangles for an over- or under-estimate, then compute the exact area using integration for comparison.

What method is used for the integral approximation based on a table?

A right-hand Riemann sum is used, using the table values corresponding to the specified interval.

How do you find the particle’s displacement and distance traveled?

Displacement is the change in position over the interval, and distance traveled is the total path length based on the position function’s values over time.

Sep 30, 2026

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