Calculate Riemann Sum

Riemann Sum is a method for approximating the area underneath a continuous function. To find the Riemann’s sum, divide the area under the curve into n equal width rectangles. Then calculate the area of each rectangle and sum the results together. This is described by the equation below:

The variable A represents the approximated area, f(xi) represents the height of each rectangle and dx (the change in x), represents the width of each rectangle. The symbol Σ means add all of the parts together.

Note: This image shows the left approximation method, see the interactive above for the alternative methods: mid-point, right, and trapezoid.

Left Riemann Sum Example

What is the left Riemann sum from 0 to 4, with n = 4, of the function:

Draw a picture and divide the area into n equal width rectangles. Set up the initial equation.

Expand the summation. Since we are using the left approximation method we are using the left point of each rectangle for the height.

Calculate the width of each rectangle.

Calculate the height of each rectangle:

First rectangle
Second rectangle.
Third rectangle.
Fourth rectangle.

Substitute widths and heights back into original equation and simplify.

The Riemann sum, or the area under the function from 0 to 4 is approximately 10. If we wanted a more accurate approximation of the area, we could increase n to a larger number.