Chapter 5.1, Problem 58E

In the following exercises, the functions $f_n$are given, where $n\ge 1$ is a natural number.

a. Find the volume of the solids S,, under the surfaces z = f(x. y) and above the region R.

b. Deteimine the limit of the volumes of the solids S as n increases without bound.

58. Use the midpoint rule with m=n to show that the average value of a function f on a rectangular region $R=\left[a,b\right]\times \left[x,d\right]$ is approximated by

   $f_{\text{ave}\approx \frac{1}{n^2}{\displaystyle \underset{j=1}{\overset{n}{\sum }}f\left(\frac{1}{2}\left(x_{i-1}++x_i\right),\frac{1}{2}\left(y_{j-1}+y_j\right)\right)}}$