Calculus: Riemann Sums and the Definite Integral Worksheet

Question 1

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

Accepted Answer

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.

Question 2

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

Accepted Answer

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.

Question 3

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

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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.

Question 7

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

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 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.

Accepted Answer

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

Question 11

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

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 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.

Accepted Answer

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.

Question 15

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

Accepted Answer

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.

Calculus: Riemann Sums and the Definite Integral

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