Riemann Midpoint Sums

This worksheet illustrates the way Riemann midpoint sums converge to the area under a curve. The rectangles associated with the area under a curve are shown in the diagram below. Use the slider to control how many rectangles are used. Observe that the height of each rectangle is the value of the function at the midpoint of the subinterval. The function is given by \[f(x)=2\sin\left(\frac{3x}2\right)+\cos\left(\frac{x}2\right).\] We are computing \[\int_0^4 f(x)\,dx\approx 1.8717011381175.\] Use the slider to change the width and number of subintervals in the Riemann sum.

Number of subintervals:


  1. What happens to the Riemann sum as you increase the number of subintervals used?
  2. Can you say about how many subintervals would be required in order for the Riemann sum to be within .01 of the actual value for the integral?
  3. You can compare this with Riemann left sums and Riemann right sums.