This applet allows you to change the two endpoints, [math]a[/math] and [math]b[/math], in the range [math]\left[0,4\right][/math], to calculate the integral [math]\displaystyle\int_a^b x^2 \; dx[/math]. The area of the integral is shown in blue. Simultaneously, a rectangle of width [math]b-a[/math] 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 [math]c[/math] is the value guaranteed by the Mean Value Theorem for Integrals. The value [math]f\left(c\right)[/math] is the average value of the function over the chosen interval and also serves as the height of the rectangle.
