Riemann Sum Practice Problems with Answers and Solutions

These Riemann sum practice problems cover left, right, midpoint and trapezoid sums, tables with uneven widths, over and under estimates, and turning a limit of sums into an integral. Try each one first. The answers come first, then full solutions.

The basic setup:

Left sums use the left end of each piece, right sums use the right end, and midpoint sums use the middle. For the ideas behind it, read the Riemann sums guide.

Problems

One function, four methods. Use f(x) = x^2 on [0, 2] with n = 4.

1. Find the left Riemann sum L_4.
2. Find the right Riemann sum R_4.
3. Find the midpoint sum M_4.
4. Find the trapezoid sum T_4, and compare all four with the exact value \int_0^2 x^2\,dx = \frac{8}{3}.

Over or under

5. Find the left sum L_4 for f(x) = \frac{1}{x} on [1, 3]. Is it an overestimate or an underestimate of \ln 3?
6. A car's velocity is measured at a few times:

Estimate the distance traveled from t = 0 to t = 6 with a left sum, a right sum and a trapezoid sum.

7. For f(x) = x^2 on [0, 2], explain why M_4 < \frac{8}{3} < T_4.

Limits of sums

8. Write \displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\frac{3}{n}\left(1 + \frac{3i}{n}\right)^2 as a definite integral and evaluate it.
9. Find the right sum R_n for f(x) = x on [0, 2] in closed form, then take the limit as n \to \infty.
10. Estimate \int_0^{\pi}\sin x\,dx with a midpoint sum, n = 4.

Answers

Step-by-step solutions

1. \Delta x = \frac{2 - 0}{4} = 0.5, so the points are 0, 0.5, 1, 1.5, 2. The left sum uses the first four:

2. The right sum uses the last four points:

3. The midpoints are 0.25, 0.75, 1.25, 1.75. Their squares are 0.0625, 0.5625, 1.5625, 3.0625, which add to 5.25:

4. With equal widths, the trapezoid sum is the average of the left and right sums:

The exact value \frac{8}{3} \approx 2.667 sits between M_4 and T_4. The midpoint sum is the closest.

5. \Delta x = 0.5 and the left points are 1, 1.5, 2, 2.5:

Since \frac{1}{x} is decreasing, the left end of each piece is the highest point, so every rectangle sticks out above the curve. It is an overestimate of \ln 3 \approx 1.099.

6. The widths are 2, 3 and 1. Multiply each width by the right height:

• Left: 2(10) + 3(14) + 1(20) = 82 m
• Right: 2(14) + 3(20) + 1(18) = 106 m
• Trapezoid: 2\cdot\frac{10+14}{2} + 3\cdot\frac{14+20}{2} + 1\cdot\frac{20+18}{2} = 24 + 51 + 19 = 94 m

7. A concave up curve bends upward, so each trapezoid's top edge lies above the curve and T_4 is too big. Each midpoint rectangle has the same area as the trapezoid under the tangent line at that midpoint, and a concave up curve lies above its tangent lines, so M_4 is too small.

8. Match the pattern \Delta x = \frac{3}{n} and x_i = 1 + \frac{3i}{n}. The interval starts at 1 and has length 3, so it is [1, 4], and the function is x^2:

9. \Delta x = \frac{2}{n} and x_i = \frac{2i}{n}. Use \sum_{i=1}^{n} i = \frac{n(n+1)}{2}:

This matches the triangle area \frac{1}{2}(2)(2) = 2.

10. \Delta x = \frac{\pi}{4}, and the midpoints are \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8}. Their sines are about 0.3827, 0.9239, 0.9239, 0.3827, which add to about 2.6131:

Since \sin x is concave down on [0, \pi], the midpoint sum overestimates the exact value 2.

Common mistakes on these problems

• Using n + 1 heights. A left or right sum with n = 4 has exactly four rectangles. List the points and cross out the one you do not use.
• Assuming equal widths in a table. In problem 6 the gaps are 2, 3 and 1. Using a single \Delta x gives the wrong answer.
• Mixing up the over and under rules. Left and right depend on increasing or decreasing. Midpoint and trapezoid depend on concavity.
• Misreading the interval in a limit. In problem 8, x_i = 1 + \frac{3i}{n} means the interval starts at 1, not 0.

Keep practicing

For unlimited sums with new functions and instant feedback, work through the Riemann sums lesson and the trapezoid sums lesson. Then see how the limit of these sums becomes an exact answer in the definite integral problems and the Fundamental Theorem problems.