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Math Grade 9-12 Answer Key

Calculus: Riemann Sums and the Definite Integral

Approximating area and connecting sums to integrals

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Calculus: Riemann Sums and the Definite Integral

Approximating area and connecting sums to integrals

Math - Grade 9-12

Instructions: Read each problem carefully. Show your setup and calculations in the space provided.
  1. 1

    Use a left Riemann sum with 4 equal subintervals to approximate the area under f(x) = x^2 on the interval [0, 4].

    For a left Riemann sum, use the x-value at the left side of each subinterval.

    The width is Δx = 1. The left endpoints are 0, 1, 2, and 3, so the sum is 1(0^2 + 1^2 + 2^2 + 3^2) = 14. The left Riemann sum is 14 square units.
  2. 2

    Use a right Riemann sum with 4 equal subintervals to approximate the area under f(x) = x^2 on the interval [0, 4].

    The width is Δx = 1. The right endpoints are 1, 2, 3, and 4, so the sum is 1(1^2 + 2^2 + 3^2 + 4^2) = 30. The right Riemann sum is 30 square units.
  3. 3

    Use a midpoint Riemann sum with 3 equal subintervals to approximate the area under f(x) = 2x + 1 on the interval [0, 6].

    First divide the interval into [0, 2], [2, 4], and [4, 6], then use the midpoint of each interval.

    The width is Δx = 2. The midpoints are 1, 3, and 5, so the sum is 2(3 + 7 + 11) = 42. The midpoint Riemann sum is 42 square units.
  4. 4

    A moving object has velocity values shown at times t = 0, 2, 4, 6, and 8 seconds: v(t) = 3, 5, 6, 4, and 2 meters per second. Use a left Riemann sum with 4 subintervals to estimate the object's displacement from t = 0 to t = 8.

    Do not use the velocity at t = 8 for a left Riemann sum on this interval.

    The time width is Δt = 2 seconds. The left velocities are 3, 5, 6, and 4, so the estimate is 2(3 + 5 + 6 + 4) = 36. The estimated displacement is 36 meters.
  5. 5

    Find the exact value of the definite integral of f(x) = 3 from x = 0 to x = 5.

    The graph forms a rectangle with width 5 and height 3. The definite integral is 5 times 3, which equals 15 square units.
  6. 6

    Find the exact value of the definite integral of f(x) = x + 2 from x = 0 to x = 3 using geometry.

    Evaluate the function at x = 0 and x = 3 to find the two vertical heights.

    The region is a trapezoid with vertical heights 2 and 5 and width 3. Its area is one half times (2 + 5) times 3, which equals 10.5. The definite integral is 10.5.
  7. 7

    Write the limit definition of the definite integral of f(x) on [a, b] using right endpoints.

    The definite integral from a to b of f(x) dx is the limit as n approaches infinity of the sum from i = 1 to n of f(a + iΔx)Δx, where Δx = (b - a)/n.
  8. 8

    Suppose f(x) is increasing and positive on [a, b]. Explain whether a left Riemann sum gives an overestimate or an underestimate, and explain whether a right Riemann sum gives an overestimate or an underestimate.

    Compare the height of each rectangle with the curve over the same subinterval.

    For an increasing positive function, a left Riemann sum gives an underestimate because each left endpoint is the lowest function value on its subinterval. A right Riemann sum gives an overestimate because each right endpoint is the highest function value on its subinterval.
  9. 9

    Use a right Riemann sum with 4 equal subintervals to approximate the area under f(x) = 4 - x on the interval [0, 4]. Then state whether the estimate is less than or greater than the exact area.

    The width is Δx = 1. The right endpoints are 1, 2, 3, and 4, giving function values 3, 2, 1, and 0. The sum is 1(3 + 2 + 1 + 0) = 6. Since the function is decreasing, the right Riemann sum is less than the exact area.
  10. 10

    A graph has 12 square units of area above the x-axis from x = 0 to x = 3 and 5 square units of area below the x-axis from x = 3 to x = 5. Find the value of the definite integral from x = 0 to x = 5.

    A definite integral represents signed area, not total area.

    Area above the x-axis counts as positive, and area below the x-axis counts as negative. The definite integral is 12 - 5 = 7.
  11. 11

    Find the average value of f(x) = x^2 on the interval [0, 3].

    The average value is 1/(3 - 0) times the integral from 0 to 3 of x^2 dx. The integral is 9, so the average value is 9/3 = 3.
  12. 12

    The sum from i = 1 to 5 of [(1 + 2i/5)^2](2/5) is a right Riemann sum. Identify the function, interval, and number of subintervals.

    Match the expression to f(a + iΔx)Δx.

    The function is f(x) = x^2. The interval is [1, 3] because the right endpoints have the form 1 + i(2/5), and the width is 2/5. The number of subintervals is 5.
  13. 13

    A graph of f has a line segment from (0, 0) to (2, 4), then a horizontal line segment from (2, 4) to (5, 4). Find the definite integral of f from x = 0 to x = 5 using geometry.

    From x = 0 to x = 2, the region is a triangle with base 2 and height 4, so its area is 4. From x = 2 to x = 5, the region is a rectangle with width 3 and height 4, so its area is 12. The definite integral is 4 + 12 = 16.
  14. 14

    Write a left Riemann sum with n subintervals for f(x) = square root of x on the interval [1, 5]. Do not evaluate the limit.

    For left endpoints, use a + (i - 1)Δx.

    The width is Δx = (5 - 1)/n = 4/n. The left endpoints are 1 + 4(i - 1)/n, so the left Riemann sum is the sum from i = 1 to n of square root of [1 + 4(i - 1)/n] times 4/n.
  15. 15

    Evaluate the definite integral of 3x^2 + 1 from x = 0 to x = 2.

    An antiderivative of 3x^2 + 1 is x^3 + x. Evaluating from 0 to 2 gives (2^3 + 2) - (0^3 + 0) = 10. The definite integral is 10.
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