Lesson I3

I3 Numerical integration with the trapezium rule Quiz: AQA Maths, Unit 9

20 questions

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Lesson I3, Numerical integration with the trapezium rule: 20 multiple choice questions for the AQA Maths (7357), Unit 9: Numerical methods, written with Revision Ninja.

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The 20 questions

  1. Which formula is the trapezium rule for n strips of width h?

    • h[y0 + y1 + ... + yn]
    • (h/2)[y0 - yn]
    • (h/2)[y0 + 2(y1 + y2 + ... + y(n-1)) + yn]
    • (h/3)[y0 + 4y1 + 2y2 + ... + yn]

  2. The trapezium rule approximates the area under a curve using which shapes?

    • Tangent lines at each point
    • Rectangles of equal height
    • Parabolas through three points
    • Trapeziums, formed by straight-line segments between the points

  3. With an interval [a, b] split into n strips, what is the strip width h?

    • h = (b - a)/n
    • h = (b + a)/n
    • h = b - a
    • h = n/(b - a)

  4. For a convex curve (concave up), what does the trapezium rule usually give?

    • An overestimate of the area
    • The exact area
    • An underestimate of the area
    • An undefined value

  5. Which statement about the accuracy of the trapezium rule is true?

    • Increasing the number of strips always increases the error
    • Increasing the number of strips generally reduces the error
    • The estimate does not depend on the number of strips
    • The estimate is always exact for any curve

  6. Use the trapezium rule with n = 2 strips on the integral from 0 to 2 of x^2 dx. What is the estimate?

    • 2.5
    • 4
    • 10/3
    • 3

  7. Use the trapezium rule with n = 2 strips on the integral from 1 to 2 of 1/x dx. What is the estimate to 4 decimal places?

    • 0.75
    • 0.7083
    • 0.6667
    • 0.6931

  8. A table gives x = 0, 1, 2, 3 with y = 1, 3, 4, 2. What is the trapezium estimate of the area under the curve?

    • 8.5
    • 10
    • 9
    • 7

  9. Use the trapezium rule with n = 4 strips on the integral from 0 to 2 of x^2 dx. What is the estimate?

    • 3
    • 2.25
    • 2.75
    • 2.5

  10. A student uses the trapezium rule with n = 2 on y = x^2 over [0, 2] and gets 3. The exact value is 8/3. How does the estimate compare?

    • The true value is greater than 3
    • The true value is exactly 3
    • The true value is 0
    • The true value 8/3 is less than 3

  11. A table gives x = 1, 2, 3, 4, 5 with y = 2, 4, 5, 3, 1. What is the trapezium estimate of the area with h = 1?

    • 14
    • 13.5
    • 15
    • 12

  12. Use the trapezium rule with n = 4 strips on the integral from 0 to 1 of x^3 dx. What is the estimate to 4 decimal places?

    • 0.2813
    • 0.2656
    • 0.3
    • 0.25

  13. In the previous trapezium estimate of the integral of x^3 from 0 to 1 with n = 4, how does the estimate compare with the true value 1/4?

    • It overestimates by 1/64
    • It is exact
    • It underestimates by 1/64
    • It overestimates by 1/8

  14. Using the trapezium rule with n = 3 strips of width 1 on [0, 3], the ordinates are y = 0, 2, 3, 0. What is the estimate?

    • 4
    • 6
    • 3
    • 5

  15. Using the trapezium rule with h = 1 on [1, 3], the ordinates are y = 2, 5, 4. What is the estimate?

    • 8
    • 6
    • 11
    • 9

  16. A student wants a more accurate trapezium rule estimate of an area. Which change is most effective?

    • Use only the two end values
    • Use a larger interval
    • Round the y-values to the nearest integer
    • Increase the number of strips, which reduces the width h

  17. Use the trapezium rule with one strip to estimate the integral from 0 to 1 of 1/(1 + x) dx. What is the estimate?

    • 1
    • 0.5
    • 0.75
    • 0.6931

  18. For a concave-down curve, what does the trapezium rule give compared with the true area?

    • An overestimate
    • The exact area
    • An undefined value
    • An underestimate

  19. Velocity values are 0, 6 and 8 m/s at t = 0, 2 and 4 s. Using the trapezium rule, what is the approximate distance travelled?

    • 28 m
    • 16 m
    • 14 m
    • 20 m

  20. Using the trapezium rule with h = 2 on [0, 4], the ordinates are y = 3, 5, 7. What is the estimate?

    • 16
    • 18
    • 24
    • 20

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