# Mean Value Theorem (Integrals) & Average Value of a Function

This applet allows you to change the two endpoints, and , in the range , to calculate the integral . The area of the integral is shown in blue. Simultaneously, a rectangle of width is created with height equal to the average value of the function over the chosen interval; that is, the area of the rectangle (in red) is exactly the same as the value of the definite integral. The value is the value guaranteed by the Mean Value Theorem for Integrals. The value is the average value of the function over the chosen interval and also serves as the height of the rectangle.