Riemann Sums and the Definite Integral as a Limit · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Areas from thinner and thinner rectangles

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. For f(x) = x^2 on [0, 2], which sum with n = 4 rectangles overshoots the true area the most?

    • The left sum, because it uses the smallest heights
    • The right sum, because the function is increasing and right endpoints take the largest heights
    • The midpoint sum, because it always overestimates
    • All three sums give the same value

  2. Compute the left Riemann sum for f(x) = x^2 on the interval [0, 2] with n = 4 rectangles.

    Answer: ______________

  3. What is a Riemann sum?

    • Rectangle areas added to estimate area under a curve
    • An exact antiderivative formula
    • A list of prime numbers

  4. Compute the right Riemann sum for f of x equals x on 0 to 4 with 4 rectangles.

    Answer: ______________

  5. Compute the midpoint Riemann sum for f(x) = x^2 on the interval [0, 2] with n = 4 rectangles.

    Answer: ______________

  6. Compute the right Riemann sum for f(x) = x^2 on the interval [0, 2] with n = 4 rectangles.

    Answer: ______________

  7. You want to estimate the area under f(x) = x^2 from 0 to 2 using n rectangles with right endpoints. Which sigma notation gives the Riemann sum?

    • The sum from i = 1 to n of (2i/n)^2 times (2/n)
    • The sum from i = 1 to n of (2i/n)^2 times (1/n)
    • The sum from i = 0 to n-1 of (2i/n)^2 times (2/n)
    • The sum from i = 1 to n of i^2 times (2/n)

  8. As n grows without bound, left, right, and midpoint sums for a continuous function approach one shared number.

    Circle one:   True   False

  9. The limit of sums of 3 plus 3i over n squared times 3 over n equals which integral?

    • Integral 3 to 6 of x squared
    • Integral 0 to 3 of x squared
    • Integral 3 to 6 of 3 x squared

  10. Which expression is the general right-endpoint Riemann sum for the integral of f(x) from a to b?

    • The sum from i = 1 to n of f(a + i(b - a)/n) times (b - a)/n
    • The sum from i = 1 to n of f(b - i(b - a)/n) times (b - a)/n
    • The sum from i = 1 to n of f(a + i) times (b - a)/n
    • The sum from i = 0 to n of f(a + i(b - a)/n) times n/(b - a)
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Answer key

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

Areas from thinner and thinner rectangles W1-mt_pwa2SRU6P9-s1

  1. The right sum, because the function is increasing and right endpoints take the largest heights · Since x^2 is increasing on [0, 2], each right endpoint is the tallest point of its subinterval, so the right sum is the biggest of the three.
  2. 1.75 · The width is 0.5 and the left endpoints 0, 0.5, 1, 1.5 give heights 0, 0.25, 1, 2.25, so the sum is 0.5 times 3.5 = 1.75.
  3. Rectangle areas added to estimate area under a curve · Heights times widths, added across pieces.
  4. 10 · Width 1 times heights 1, 2, 3, 4 gives 10.
  5. 2.625 · The midpoints 0.25, 0.75, 1.25, 1.75 give heights 0.0625, 0.5625, 1.5625, 3.0625, and 0.5 times their total 5.25 is 2.625.
  6. 3.75 · The width is 0.5 and the right endpoints 0.5, 1, 1.5, 2 give heights 0.25, 1, 2.25, 4, so the sum is 0.5 times 7.5 = 3.75.
  7. The sum from i = 1 to n of (2i/n)^2 times (2/n) · With a = 0 and b = 2, the width of each rectangle is (2 - 0)/n = 2/n, the right endpoint of rectangle i is 2i/n, and the rectangle area is height times width.
  8. True · Thin pieces erase sample differences, squeezing all three to the integral.
  9. Integral 3 to 6 of x squared · Width 3 over n means length 3 from a equals 3, with square heights.
  10. The sum from i = 1 to n of f(a + i(b - a)/n) times (b - a)/n · The width of each piece is (b - a)/n, and the right endpoint of piece i is a plus i times that width, giving the sum of f(a + i(b - a)/n) times (b - a)/n.
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