Numerical Quadrature: Newton-Cotes and Adaptive Rules · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Area under the curve, panel by panel

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. Use Simpson's rule with one panel to approximate the integral of x cubed from 0 to 2. What value do you get?

    Answer: ______________

  2. Use the single panel trapezium rule to approximate the integral of x squared from 0 to 2.

    • 4
    • 8
    • 2
    • 8/3

  3. Use one trapezium panel for x squared from 0 to 2. What do you get?

    • 2
    • 4
    • 8

  4. An adaptive rule refines every subinterval by the same amount.

    Circle one:   True   False

  5. Simpson's rule has error order 4. An estimate has error 0.48. After halving the step, what is the predicted new error?

    Answer: ______________

  6. Simpson rule comes from integrating which interpolant exactly?

    • The constant equal to the midpoint value
    • The line through the two endpoints
    • The parabola through the endpoints and the midpoint

  7. The trapezium rule has error order 2. If the step size is halved, what happens to the error?

    • It is halved
    • It is divided by 4
    • It is divided by 8
    • It stays the same

  8. An adaptive rule estimates the error on each subinterval and subdivides the subinterval with the largest estimate first. True or false?

    Circle one:   True   False

  9. Apply the composite trapezium rule with 2 panels to x squared on [0, 2]. What value do you get?

    Answer: ______________

  10. A fixed budget of function evaluations is available. When does an adaptive rule beat a uniform mesh?

    • The integrand is a low degree polynomial
    • The integrand is constant on the interval
    • The integrand is smooth almost everywhere but has a sharp peak in one small region
    • The local errors are already evenly spread
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Area under the curve, panel by panel W1-mt_fT5kRCUiD5-s1

  1. 4 · Simpson gives (2/6) times (0 plus 4 plus 8), which is 4, the exact integral.
  2. 4 · One panel of width 2 gives 2 times (0 plus 4) divided by 2, which is 4.
  3. 4 · Width 2 times the average of 0 and 4 is 4.
  4. False · It subdivides only the subintervals with the largest error estimates.
  5. 0.03 · Order 4 divides the error by 2 to the power 4, which is 16, and 0.48 divided by 16 is 0.03.
  6. The parabola through the endpoints and the midpoint · The midpoint parabola is the curve Simpson integrates.
  7. It is divided by 4 · Error order 2 means the error scales with h squared, so halving h quarters it.
  8. True · True. That greedy choice concentrates work where the integrand is hardest.
  9. 3 · With h = 1 the sum is 0 plus 4 over 2, plus 1, which is 3.
  10. The integrand is smooth almost everywhere but has a sharp peak in one small region · Adaptivity wins when difficulty is localised, so work can move to the peak and away from calm regions.
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