Lesson H1-H3

H1-H3 Fundamental Theorem, definite integrals and areas Quiz: AQA Maths, Unit 8

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson H1-H3, Fundamental Theorem, definite integrals and areas: 20 multiple choice questions for the AQA Maths (7357), Unit 8: Integration, written with Revision Ninja.

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The 20 questions

  1. The Fundamental Theorem of Calculus links integration and differentiation. If F'(x) = f(x), what is the integral of f(x) from a to b?

    • F(b) + F(a)
    • F(a) - F(b)
    • F'(b) - F'(a)
    • F(b) - F(a)

  2. If F(x) is the integral from a to x of f(t) dt, what is dF/dx?

    • f(x)
    • x f(x)
    • F(x) - F(a)
    • f(a)

  3. Evaluate the integral from 1 to 3 of 2x dx.

    • 8
    • 9
    • 4
    • 6

  4. Evaluate the integral from 0 to pi of sin x dx.

    • pi
    • 0
    • -2
    • 2

  5. Evaluate the integral from 0 to 1 of e^(2x) dx.

    • 2e^2 - 2
    • e^2 - 1
    • e^2/2
    • (e^2 - 1)/2

  6. Which statement is true about the integral of f(x) from a to b when f(x) is negative on the whole interval [a, b]?

    • The value is always zero
    • The value of the integral is positive
    • The value equals the length b - a
    • The value of the integral is negative

  7. The area between the curves y = f(x) (above) and y = g(x) (below) for a <= x <= b is found by which expression?

    • integral from a to b of (f(x) + g(x)) dx
    • integral from a to b of (f(x) - g(x)) dx
    • integral from a to b of (g(x) - f(x)) dx
    • integral from a to b of f(x) dx + integral of g(x) dx

  8. Find the area between y = 4 - x^2 and the x-axis for 0 <= x <= 2.

    • 16/3
    • 8/3
    • 32/3
    • 8

  9. Find the area enclosed between y = x and y = x^2 for 0 <= x <= 1.

    • 1/3
    • 5/6
    • 1/6
    • 1/2

  10. For what positive value of k is the integral from 0 to k of 3x^2 dx equal to 8?

    • 4
    • 8
    • 3
    • 2

  11. Evaluate the integral from 1 to 4 of 1/x dx.

    • ln 3, approximately 1.099
    • ln 4, approximately 1.386
    • 3
    • 3/4, approximately 0.75

  12. A particle has velocity v = 3t^2 m/s. What is its displacement change from t = 1 s to t = 2 s?

    • 7 m
    • 3 m
    • 8 m
    • 9 m

  13. Find the area between y = e^x and y = 1 for 0 <= x <= 1.

    • e - 2
    • e - 1
    • 2 - e
    • e

  14. Which statement about a definite integral from a to b of f(x) dx is correct?

    • Its value does not depend on the letter used for the variable of integration
    • It is always positive
    • It always equals the area under the curve even when f is negative
    • It always equals f(b) - f(a)

  15. Find the area enclosed between the curve y = x(x - 2) and the x-axis.

    • 2
    • 4/3
    • 8/3
    • -4/3

  16. Find the area enclosed between y = x^2 and y = 2x - x^2.

    • 1/6
    • 2/3
    • 1/3
    • 1

  17. Find the value of a, with a not equal to 2, such that the integral from a to 2 of 2x dx equals zero.

    • a = -2
    • a = 0
    • a = -4
    • a = 2

  18. Let F(x) = integral from 0 to x of (t^2 - 1) dt. Which values of x give stationary points of F?

    • x = 1 only
    • x = 0 and x = 1
    • x = -1 and x = 1
    • x = 0 only

  19. The area under y = 1/x^2 from x = 1 to x = b equals 1/2. What is b?

    • 2
    • sqrt(2)
    • 1/2
    • 4

  20. A curve y = f(x) lies above the x-axis for 0 <= x <= 4, and the integral of f from 0 to 4 is 10. What is the area between the curve and the x-axis?

    • 4
    • 40
    • -10
    • 10

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