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Riemann Sums Practice Problems

Find Δx, list the sample points, evaluate the function and multiply by the width. Type numbers as decimals or fractions; answers rounded to three decimal places are accepted where the exact value isn't a whole number.

  1. 1.

    The interval is divided into subintervals of equal width. What is ?

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    .

  2. 2.

    Find the left Riemann sum for on .

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    , and the left endpoints are 0, 1, 2 and 3. .

  3. 3.

    Find the right Riemann sum for on .

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    The right endpoints are 1, 2, 3 and 4. .

  4. 4.

    Find the midpoint Riemann sum for on .

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    The midpoints are 0.5, 1.5, 2.5 and 3.5. . For a straight line the midpoint sum is exact: the region is a trapezoid with area .

  5. 5.

    Find the right Riemann sum for on .

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    , and the right endpoints are 1.5, 2, 2.5 and 3, with -values 3.25, 5, 7.25 and 10. .

  6. 6.

    Find the left Riemann sum for on .

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    , and the left endpoints are 1, 1.5, 2 and 2.5. .

  7. 7.

    Find the midpoint Riemann sum for on .

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    , and the midpoints are and , where . . The exact area is 2, so the midpoint sum overestimates, as expected for a concave-down function.

  8. 8.

    A car's velocity is measured as , , and m/s. Use a left Riemann sum to estimate the distance traveled from to seconds.

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    The subintervals have widths 2, 3 and 4. Using the left values: m.

  9. 9.

    A car's velocity is measured as , , and m/s. Use a right Riemann sum to estimate the distance traveled from to seconds.

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    Using the right values: m.

  10. 10.

    A car's velocity is measured as , , and m/s. Use a trapezoidal sum to estimate the distance traveled from to seconds.

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    Each trapezoid has area (average of the two heights) × width: m. That's also the average of the left and right sums, .

  11. 11.

    is positive and decreasing on . Is the left Riemann sum an overestimate or an underestimate of the area under ?

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    For a decreasing function, the left endpoint is the highest point of each subinterval, so every rectangle sticks out above the curve. The left sum is an overestimate.

  12. 12.

    Find the exact area under on by evaluating .

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    The sum is . As , this approaches .
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MPhil in Mathematics, Founder of Mathovia
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Riemann Sums Practice Problems

These problems follow the Riemann sums lesson. Each one asks for a left, right, midpoint or trapezoidal estimate of an area, or for the exact area as a limit of sums.

The method

  1. Find , or read the widths from the table.
  2. List the sample points: left endpoints, right endpoints or midpoints.
  3. Evaluate at each sample point.
  4. Multiply each value by its width and add.

Common mistakes to watch for

  • Using heights. rectangles need heights.
  • Using endpoints for a midpoint sum. Find the midpoints first.
  • Assuming equal widths in a table. Check the gaps between the -values.

Frequently Asked Questions

Do I need to round my answers?+

No. Exact answers such as 77/60 or pi*sqrt(2)/2 are accepted, and so are decimals rounded to three places, such as 1.283 or 2.221.

What if the measurements are unevenly spaced?+

Multiply each height by the width of its own subinterval. In the velocity problems here, the widths are 2, 3 and 4 seconds.

More practice problems

Antiderivatives Practice Problems Limits at Infinity Practice Problems Concavity and Inflection Points Practice Problems