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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 ∫032x dx\int_0^3 2x\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 9.

  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^2, and 3^2-0^2=9.
  3. Check bounds and interpretation: The correct answer is 9.

An antiderivative is x^2, and 3^2-0^2=9.

2. QuestionMultiple choice

What is ∫143 dx\int_1^4 3\,dx?

Please select an answer first.

Correct answer: B

The correct answer is 9.

  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: The rectangle has height 3 and width 3, so its area is 9.
  3. Check bounds and interpretation: The correct answer is 9.

The rectangle has height 3 and width 3, so its area is 9.

3. QuestionMultiple choice

What is ∫−11x3 dx\int_{-1}^1 x^3\,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: x^3 is odd and the interval is symmetric about zero.
  3. Check bounds and interpretation: The correct answer is 0.

x^3 is odd and the interval is symmetric about zero.

4. QuestionMultiple choice

What is ∫02(x+1) dx\int_0^2 (x+1)\,dx?

Please select an answer first.

Correct answer: D

The correct answer is 4.

  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^2/2+x; evaluate at 2 to get 4.
  3. Check bounds and interpretation: The correct answer is 4.

An antiderivative is x^2/2+x; evaluate at 2 to get 4.

5. QuestionMultiple choice

If f(x)≥0 on [a,b], what does ∫abf(x) dx\int_a^b f(x)\,dx represent?

Please select an answer first.

Correct answer: A

The correct answer is The area under the graph.

  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: When the graph is nonnegative, net signed area equals ordinary geometric area.
  3. Check bounds and interpretation: The correct answer is The area under the graph.

When the graph is nonnegative, net signed area equals ordinary geometric area.

6. QuestionMultiple choice

What is ∫13(4x−2) dx\int_1^3 (4x-2)\,dx?

Please select an answer first.

Correct answer: B

The correct answer is 12.

  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 2x^2-2x; values are 12 and 0, giving 12.
  3. Check bounds and interpretation: The correct answer is 12.

An antiderivative is 2x^2-2x; values are 12 and 0, giving 12.

7. QuestionMultiple choice

What is the net signed area if 5 square units lie above the x-axis and 2 below?

Please select an answer first.

Correct answer: C

The correct answer is 3 square units.

  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: Signed area is 5-2=3.
  3. Check bounds and interpretation: The correct answer is 3 square units.

Signed area is 5-2=3.

8. QuestionMultiple choice

What is ∫0πsin⁡x dx\int_0^\pi \sin x\,dx?

Please select an answer first.

Correct answer: D

The correct answer is 2.

  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 -cos x, giving -cosπ+cos0=2.
  3. Check bounds and interpretation: The correct answer is 2.

An antiderivative is -cos x, giving -cosπ+cos0=2.

9. QuestionMultiple choice

What is ∫016x2 dx\int_0^1 6x^2\,dx?

Please select an answer first.

Correct answer: A

The correct answer is 2.

  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 2x^3, which evaluates to 2.
  3. Check bounds and interpretation: The correct answer is 2.

An antiderivative is 2x^3, which evaluates to 2.

10. QuestionMultiple choice

What does reversing the bounds do to a definite integral?

Please select an answer first.

Correct answer: B

The correct answer is It changes the sign.

  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: The property is ∫_b^a f(x)dx=-∫_a^b f(x)dx.
  3. Check bounds and interpretation: The correct answer is It changes the sign.

The property is ∫_b^a f(x)dx=-∫_a^b f(x)dx.

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