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AP Calculus AB Unit 6B: Definite Integration & Riemann Sums — practice questions — page 3

Explore 70 free Unit 6B: Definite Integration & Riemann Sums questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 6B: Definite Integration & Riemann Sums questions and exercises with answers

Showing questions 61–70 of 70.

∫ x sin x dx (integration by parts):Read the answer and worked solution →∫ sec²x dx =Read the answer and worked solution →Area under y = x² from x = 0 to x = 2:Read the answer and worked solution →∫ from 0 to ∞ e^(-x) dx =Read the answer and worked solution →∫ x ln x dx (integration by parts: u = ln x, dv = x dx):Read the answer and worked solution →∫ from 1 to e (1/x) dx =Read the answer and worked solution →Reduction formula example: ∫ sinⁿ x dx (for n ≥ 2) =Read the answer and worked solution →∫ from 0 to 2 (x² - 2x) dx =Read the answer and worked solution →For ∫ dx / sqrt(a² - x²), let x = a sin θ. Result:Read the answer and worked solution →∫ from 0 to π/2 sin x cos x dx =Read the answer and worked solution →

Worked practice questions

∫ x sin x dx (integration by parts):

  1. sin x - x cos x + C
  2. -x cos x + sin x + C
  3. x cos x
  4. cos x
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Correct answer: B. -x cos x + sin x + C

u = x, du = dx; dv = sin x dx, v = -cos x. We have ∫ x sin x dx = -x cos x + ∫ cos x. So dx = -x cos x + sin x + C.

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∫ sec²x dx =

  1. tan x + C
  2. sin x
  3. ln|sec x|
  4. cot x
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Correct answer: A. tan x + C

d/dx(tan x) = sec²x, so ∫ sec²x dx = tan x + C. (And ∫ csc²x dx = -cot x + C, ∫ sec x tan x dx = sec x + C, ∫ csc x cot x dx = -csc x + C.)

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Area under y = x² from x = 0 to x = 2:

  1. 4
  2. 8/3
  3. 2
  4. 1
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Correct answer: B. 8/3

Area = ∫₀² x² dx = [x³/3]₀² = 8/3.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits