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

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 1–30 of 70.

The value of $\int_a^a f(x)\,dx$ is:Read the answer and worked solution →For a definite integral with limits swapped, which identity holds?Read the answer and worked solution →For $a < c < b$, the integral $\int_a^b f(x)\,dx$ equals:Read the answer and worked solution →If $f(x)$ is an **odd** function, then $\int_{-a}^{a} f(x)\,dx$ equals:Read the answer and worked solution →If $f(x)$ is an **even** function, then $\int_{-a}^{a} f(x)\,dx$ equals:Read the answer and worked solution →The 'King's property' of definite integration is:Read the answer and worked solution →$\int_1^3 x\,dx$ equals:Read the answer and worked solution →$\int_0^1 \dfrac{dx}{1+x^2}$ equals:Read the answer and worked solution →$\int_{-\pi/2}^{\pi/2} \sin^3 x\,dx$ equals:Read the answer and worked solution →$\int_{\pi/6}^{\pi/3} \cos x\,dx$ equals:Read the answer and worked solution →$\int_{-1}^{5}(2x+3)\,dx$ equals:Read the answer and worked solution →$\int_0^2 x^2\,dx$ equals:Read the answer and worked solution →$\int_{-1}^{1} \dfrac{x^2}{1+x^2}\,dx$ equals:Read the answer and worked solution →$\int_0^{\pi/2} \cos^3 x\,dx$ equals:Read the answer and worked solution →$\int_0^{\pi/2} \sin x\,dx$ equals:Read the answer and worked solution →Using the King's property, $\int_0^a f(x)\,dx$ equals:Read the answer and worked solution →$\int_0^{\pi/2} \sin^2 x\,dx$ equals:Read the answer and worked solution →If $f(x) = x^3 \sin^4 x$, then $\int_{-\pi/4}^{\pi/4} f(x)\,dx$ equals:Read the answer and worked solution →$\int_{-2}^{2} (x^3 + x^2)\,dx$ equals:Read the answer and worked solution →$\int_0^1 e^x\,dx$ equals:Read the answer and worked solution →$\int_0^4 (x - x^2)\,dx$ equals:Read the answer and worked solution →Evaluate $\int_2^3 7^x\,dx$:Read the answer and worked solution →$\int_{-1}^{1} |5x - 3|\,dx$ equals:Read the answer and worked solution →$\int_1^2 \dfrac{\log x}{x^2}\,dx$ equals:Read the answer and worked solution →Using the King's property, evaluate $\int_0^{\pi/2} \dfrac{dx}{1 + \sqrt[3]{\tan x}}$:Read the answer and worked solution →$\int_0^{\pi/2} \sqrt{1 - \cos 4x}\,dx$ equals:Read the answer and worked solution →$\int_0^{\pi/2} \cos^2 x\,dx$ equals:Read the answer and worked solution →Using definite integration as the limit of a sum, $\int_1^2 (2x + 5)\,dx$ equals:Read the answer and worked solution →Evaluate $\displaystyle\lim_{n\to\infty} \sum_{r=1}^{n} \dfrac{1}{n + r}$Read the answer and worked solution →$\displaystyle\int_0^1 x^2(1-x)^3\, dx = ?$ (Beta function)Read the answer and worked solution →

Worked practice questions

The value of $\int_a^a f(x)\,dx$ is:

  1. $f(a)$
  2. $0$
  3. $2f(a)$
  4. Undefined
Show answer and solution

Correct answer: B. $0$

By the Fundamental Theorem, $\int_a^a f(x)\,dx = g(a) - g(a) = 0$ for any antiderivative $g$. The interval has zero width, so the signed area is zero.

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For a definite integral with limits swapped, which identity holds?

  1. $\int_a^b f\,dx = \int_b^a f\,dx$
  2. $\int_a^b f\,dx = -\int_b^a f\,dx$
  3. $\int_a^b f\,dx = \int_b^a f\,dx + c$
  4. $\int_a^b f\,dx = 1/\int_b^a f\,dx$
Show answer and solution

Correct answer: B. $\int_a^b f\,dx = -\int_b^a f\,dx$

**Property II:** swapping limits flips the sign. $\int_a^b f\,dx = g(b)-g(a) = -[g(a)-g(b)] = -\int_b^a f\,dx$.

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For $a < c < b$, the integral $\int_a^b f(x)\,dx$ equals:

  1. $\int_a^c f(x)\,dx - \int_c^b f(x)\,dx$
  2. $\int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
  3. $\int_a^c f(x)\,dx \cdot \int_c^b f(x)\,dx$
  4. $\int_c^b f(x)\,dx - \int_a^c f(x)\,dx$
Show answer and solution

Correct answer: B. $\int_a^c f(x)\,dx + \int_c^b f(x)\,dx$

Property IV (additivity): the integral can be split at any interior point $c$: $\int_a^b = \int_a^c + \int_c^b$.

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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