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Integrals Practice Tests

These Grade 12 Math practice tests develop integration skills through antiderivatives, definite integrals, substitution, accumulation functions, average value, area between curves, motion, Riemann sums, and the Fundamental Theorem of Calculus.

Skills covered

  • Finding antiderivatives of polynomial, trigonometric, exponential, logarithmic, and composite functions
  • Evaluating definite integrals and interpreting signed area, total change, and units
  • Using substitution, symmetry, and the Fundamental Theorem of Calculus
  • Calculating average values, areas between curves, and accumulated quantities
  • Applying integrals to velocity, flow, distance, Riemann sums, and real-world rates

How to use these practice tests

First identify whether the integral represents an antiderivative, a signed area, or accumulated change. Then select the right rule, substitution, bounds, and units and check the interpretation.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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Grade 12 Integrals Practice Tests

Practice Grade 12 Math questions about antiderivatives, indefinite and definite integrals, substitution, area, accumulation functions, average value, motion, Riemann sums, and the Fundamental Theorem of Calculus.

Choose the best answer. Identify whether the task asks for an antiderivative, accumulation, area, average value, or total change, then apply the appropriate bounds and rule.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What is ∫02∣x−1∣ dx\int_0^2 |x-1|\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 1.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Split at x=1; two triangles each have area 1/2.
  3. Check bounds and interpretation: The correct answer is 1.

Split at x=1; two triangles each have area 1/2.

2. QuestionMultiple choice

What is ∫−22(x2+1) dx\int_{-2}^2 (x^2+1)\,dx?

Please select an answer first.

Correct answer: B

The correct answer is 283\frac{28}{3}.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: An antiderivative is x^3/3+x; evaluate at ±2 to obtain 28/3.
  3. Check bounds and interpretation: The correct answer is 283\frac{28}{3}.

An antiderivative is x^3/3+x; evaluate at ±2 to obtain 28/3.

3. QuestionMultiple choice

If f is odd, what is ∫−aaf(x) dx\int_{-a}^a f(x)\,dx?

Please select an answer first.

Correct answer: C

The correct answer is 0.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Odd-function values cancel on a symmetric interval.
  3. Check bounds and interpretation: The correct answer is 0.

Odd-function values cancel on a symmetric interval.

4. QuestionMultiple choice

If f is even, how can ∫−aaf(x) dx\int_{-a}^a f(x)\,dx be simplified?

Please select an answer first.

Correct answer: D

The correct answer is 2∫0af(x) dx2\int_0^a f(x)\,dx.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Even-function areas on the two sides are equal.
  3. Check bounds and interpretation: The correct answer is 2∫0af(x) dx2\int_0^a f(x)\,dx.

Even-function areas on the two sides are equal.

5. QuestionMultiple choice

What is the total distance traveled if velocity changes sign?

Please select an answer first.

Correct answer: A

The correct answer is Integrate speed or split the interval where velocity is zero.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: A signed velocity integral gives displacement; distance requires nonnegative speed.
  3. Check bounds and interpretation: The correct answer is Integrate speed or split the interval where velocity is zero.

A signed velocity integral gives displacement; distance requires nonnegative speed.

6. QuestionMultiple choice

If F(x)=∫0xf(t) dtF(x)=\int_0^x f(t)\,dt and f changes from positive to negative at c, what may happen at c?

Please select an answer first.

Correct answer: B

The correct answer is F may have a local maximum.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: F'=f, so F changes from increasing to decreasing.
  3. Check bounds and interpretation: The correct answer is F may have a local maximum.

F'=f, so F changes from increasing to decreasing.

7. QuestionMultiple choice

What is the integral of a rate with units gallons per minute over minutes?

Please select an answer first.

Correct answer: C

The correct answer is Gallons.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Multiplying rate units by time leaves the accumulated quantity.
  3. Check bounds and interpretation: The correct answer is Gallons.

Multiplying rate units by time leaves the accumulated quantity.

8. QuestionMultiple choice

What does a Riemann sum approximate?

Please select an answer first.

Correct answer: D

The correct answer is A definite integral.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: A Riemann sum adds rectangle contributions to approximate accumulated area.
  3. Check bounds and interpretation: The correct answer is A definite integral.

A Riemann sum adds rectangle contributions to approximate accumulated area.

9. QuestionMultiple choice

In a left Riemann sum for an increasing function, the rectangles usually:

Please select an answer first.

Correct answer: A

The correct answer is Underestimate the area.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Left endpoints give heights below later values when the function is increasing.
  3. Check bounds and interpretation: The correct answer is Underestimate the area.

Left endpoints give heights below later values when the function is increasing.

10. QuestionMultiple choice

What happens to a Riemann-sum approximation as the partition gets finer?

Please select an answer first.

Correct answer: B

The correct answer is It generally approaches the definite integral.

  1. Identify the integral structure: Determine whether the question asks for an antiderivative, a definite accumulation, area, average value, substitution, or a calculus application.
  2. Apply the integration method: Smaller subintervals improve the approximation under standard integrability conditions.
  3. Check bounds and interpretation: The correct answer is It generally approaches the definite integral.

Smaller subintervals improve the approximation under standard integrability conditions.

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