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 \int_a^b f(x)\,dx by dividing the interval [a, b] 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.
Example 1
easyApproximate \int_0^2 x^2\,dx using a left Riemann sum with n = 4 equal subintervals.
Example 2
mediumApproximate \int_1^3 \frac{1}{x}\,dx using a right Riemann sum with n = 4 subintervals and classify the estimate.
Example 3
easyUse a midpoint sum with n=2 to approximate \int_0^4 (x+1)\,dx.
Example 4
hardWrite the right Riemann sum for \int_0^1 x^3\,dx with n subintervals as a sigma expression and evaluate the limit.