The average value of a nonnegative continuous function
This interactive figure illustrates the approximation of the area under a curve using the left endpoints, right endpoints, and midpoints when computed using rectangles.
This interactive figure also illustrates an approximate value for the average value of a continuous function over the interval . If a single, large rectangle is constructed on the -axis with width and with area equal to the area of the rectangles, then the height of that rectangle is the average value of the heights of the rectangles. As increases to infinity, the height of the single large rectangle becomes a good approximation for the average value of the continuous function on the interval .
(Why do we require the function to be nonnegative? What problems arise if you drag the function so that parts of it lie below the -axis?)
Developed for use with Thomas' Calculus, published by Pearson.