Practice Riemann Sums in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A method of approximating the definite integral by dividing the interval into subintervals and summing the areas of rectangles (or trapezoids) whose heights are determined by the function.
Imagine filling the area under a curve with thin rectangles. The more rectangles you use, the better the approximation. In the limit of infinitely many infinitely thin rectangles, you get the exact area—which is the definite integral.
Showing a random 20 of 50 problems.
Example 1
easyWhat does a Riemann sum approximate as ?
Example 2
mediumEstimate using a left Riemann sum with .
Example 3
easyAs , approaches what value?
Example 4
mediumSame table as the previous question. Find the right Riemann sum for with .
Example 5
easyFor with , list the right endpoints.
Example 6
challengeFor , show the limit of right Riemann sums equals 1.
Example 7
mediumEstimate using a right Riemann sum with .
Example 8
hardExpress as the limit of a right Riemann sum, then evaluate.
Example 9
easyUse a midpoint sum with to approximate .
Example 10
mediumApproximate using a midpoint sum with .
Example 11
mediumFor an increasing function on , which Riemann sum is an upper bound: left, right, or midpoint?
Example 12
mediumEstimate using the trapezoidal rule with .
Example 13
easyEstimate with a left sum, .
Example 14
mediumFor , , find the midpoint sum (3 decimal places).
Example 15
mediumEstimate using a midpoint sum with .
Example 16
easyApproximate using a left Riemann sum with equal subintervals.
Example 17
hardFor , evaluate the right Riemann sum as a function of and take the limit.
Example 18
mediumApproximate using a right Riemann sum with subintervals and classify the estimate.
Example 19
easyFor with subintervals, what is ?
Example 20
challengeEvaluate from the definition using the limit of right Riemann sums.