The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. 
Home Applications of the Integral Averages Average Value  
Search the VIAS Library  Index  
Average Value
Given n numbers y_{l}, ..., y_{n}, their average value is defined as . If all the y_{i} are replaced by the average value y_{ave}, the sum will be unchanged, y_{1} + ... + y_{n} = y_{ave} + ... + y_{ave} = ny_{ave}. If f is a continuous function on a closed interval [a, b], what is meant by the average value of f between a and b (Figure 6.5.1)? Figure 6.5.1 Let us try to imitate the procedure for finding the average of n numbers. Take an infinite hyperreal number H and divide the interval [a, b] into infinitesimal subintervals of length dx = (b  a)/H. Let us "sample" the value of f at the H points a, a + dx, a + 2 dx, ..., a + (H  1) dx. Then the average value of f should be infinitely close to the sum of the values of f at a, a + dx, ..., a + (H  1) dx, divided by H. Thus Since ,
and we have Taking standard parts, we are led to DEFINITION Let f be continuous on [a, b]. The average value of f between a and b is Geometrically, the area under the curve y = f(x) is equal to the area under the constant curve y = f_{ave} between a and b,


Home Applications of the Integral Averages Average Value 