Lesson H4

H4 Integration as a limit of a sum Quiz: AQA Maths, Unit 8

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson H4, Integration as a limit of a sum: 20 multiple choice questions for the AQA Maths (7357), Unit 8: Integration, written with Revision Ninja.

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The 20 questions

  1. In the limit of a sum, what does the definite integral of f from a to b represent?

    • The sum of f(x_i) with no width term
    • The sum of f(x_i) squared times the strip width
    • The limit as n tends to infinity of the sum of f(x_i) times the strip width
    • The sum of x_i divided by the strip width

  2. Using n rectangles of equal width delta x = (b - a)/n, the sum of f(x_i) delta x tends to what as n tends to infinity?

    • The average gradient of f
    • f(b) - f(a)
    • The derivative of f at a
    • The definite integral of f from a to b

  3. An interval [0, 2] is split into 4 equal strips. What is the width of each strip?

    • 2
    • 0.4
    • 0.5
    • 0.25

  4. Which definite integral is the limit of the sum of (r/n)^2 times (1/n) for r = 1 to n as n tends to infinity?

    • integral from 0 to 1 of x^2 dx
    • integral from 0 to n of x^2 dx
    • integral from 0 to 1 of x dx
    • integral from 1 to 2 of x^2 dx

  5. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of (r/n)^2?

    • 2/3
    • 1/3
    • 1
    • 1/2

  6. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of (1 + r/n)?

    • 1
    • 1/2
    • 3/2
    • 2

  7. In the notation integral f(x) dx, what does dx represent in the limit of a sum?

    • A fixed width of 1
    • The height of each rectangle
    • The derivative of f
    • An infinitesimally small strip width, with delta x tending to zero

  8. Using right endpoints, 2 rectangles of width 1 approximate the area under y = x^2 on [0, 2]. What is the total area of the rectangles?

    • 3
    • 8
    • 5
    • 4

  9. Using left endpoints, 2 rectangles of width 1 approximate the area under y = x^2 on [0, 2]. What is the total area of the rectangles?

    • 5
    • 4
    • 1
    • 2

  10. What is the limit as n tends to infinity of the sum from r = 1 to n of (2r/n)(2/n)?

    • 4
    • 2
    • 8
    • 1

  11. For the integral from 1 to 5 of f(x) dx split into n = 4 equal strips, what is the strip width?

    • 2
    • 4
    • 0.25
    • 1

  12. What is the limit as n tends to infinity of (1/n^2) times the sum from r = 1 to n of r?

    • 1/3
    • 1
    • 1/2
    • 2

  13. What is the limit as n tends to infinity of the sum from r = 1 to n of (3 + 2r/n)(2/n)?

    • 8
    • 6
    • 4
    • 10

  14. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of e^(r/n)?

    • e + 1
    • e - 1
    • 1/e
    • e

  15. Using n equal strips and right endpoints for the integral from 0 to 3 of x dx, what is the Riemann sum?

    • (9/2)(1 - 1/n)
    • 3(1 + 1/n)
    • (9/2)(1 + 1/n)
    • (9/2)(1 + 1/n^2)

  16. What is the limit as n tends to infinity of the sum from r = 1 to n of 1/(n + r)?

    • ln 3
    • 1/2
    • 1
    • ln 2

  17. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of (r/n)^3?

    • 1
    • 1/4
    • 1/3
    • 1/2

  18. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of sin(pi r/n)?

    • 0
    • 1
    • 2/pi
    • pi/2

  19. Which definite integral is the limit of the sum from r = 1 to n of (1 + 2r/n)(2/n) as n tends to infinity?

    • integral from 0 to 1 of (1 + x) dx
    • integral from 0 to 2 of (1 + x) dx
    • integral from 0 to 2 of x dx
    • integral from 1 to 2 of (1 + x) dx

  20. What is the limit as n tends to infinity of (1/n) times the sum from r = 1 to n of sqrt(r/n)?

    • 2/3
    • 1
    • 1/2
    • 3/2

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