Chapter 5.1, Problem 52E

In the following exercises, the function f is given in terms of double integrals.

a. Determine the explicit form of the function f.

b. Find the volume of the solid tinder the surface z = f(x. v) and above the region R.

C. Find the average value of the function f on R.

d. Use a computer algebra system (CAS) to plot z = f(x. y) and z = f_avein the same system of coordinates.

52. Show that ${\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}y\left(x\right)+g\left(y\right)dx=\frac{1}{2}\left(d^2-c^2\right)\left({\displaystyle \underset{0}{\overset{b}{\int }}f\left(x\right)dx}\right)+\frac{1}{2}}\left(b^2-a^2\left({\displaystyle \underset{c}{\overset{d}{\int }}g\left(y\right)dy}\right)\right)}$