AP Calculus AB — free MCQ practice

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AP Calculus AB subjects

Mathematics (1,198)

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Unit 3B: Implicit Differentiation and Inverse Functions Unit 3A: Chain Rule and Composite Functions Unit 1A3: Function Composition and Inverses Unit 1A1: Relations, Functions and Graphs Unit 1A2: Trigonometric Functions & Inverse Trig Unit 2: Differentiation and Fundamental Derivative Rules Unit 6A: Indefinite Integrals & Fundamental Theorem Unit 1B3: Asymptotic Behavior and Infinite Limits Unit 1B1: Limit Concepts and Graphical Limits Unit 1B2: Algebraic Limits and Continuity Unit 6B: Definite Integration & Riemann Sums Unit 8A: Areas Between Curves Unit 8B: Volumes of Revolution and Cross-Sections Unit 7: Differential Equations and Modeling Unit 4-5: Applications of Derivatives & Curve Sketching

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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 →

Worked practice questions

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

$f(a)$
$0$
$2f(a)$
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?

$\int_a^b f\,dx = \int_b^a f\,dx$
$\int_a^b f\,dx = -\int_b^a f\,dx$
$\int_a^b f\,dx = \int_b^a f\,dx + c$
$\int_a^b f\,dx = 1/\int_b^a f\,dx$

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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:

$\int_a^c f(x)\,dx - \int_c^b f(x)\,dx$
$\int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
$\int_a^c f(x)\,dx \cdot \int_c^b f(x)\,dx$
$\int_c^b f(x)\,dx - \int_a^c f(x)\,dx$

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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