Chapter 6.6, Problem 38E

a.

To determine

To show: The surface area of the ellipsoid is $S=\frac{4\pi b}{a}{\displaystyle \underset{a}{\overset{b}{\int }}\sqrt{a^2-c^2{x}^2}}dx$.

b.

To determine

To show: The surface area of the ellipsoid in terms of u is $S=\frac{4\pi b}{\sqrt{a^2-b^2}}{\displaystyle \underset{0}{\overset{\sqrt{a^2-b^2}}{\int }}\sqrt{a^2-u^2}}du$ obtained in part a with $u=cx$

c.

To determine

To show: The surface area of the ellipsoid is $S=2\pi b\left(b+\frac{a^2}{\sqrt{a^2-b^2}}{\sin }^{-1}\frac{\sqrt{a^2-b^2}}{a}\right)$.

d.

To determine

To find: The units of the surface area obtained in part c. when the units of a and b is in meters.

e.

To determine

To show: The surface area obtained in part a. is $S=4\pi a^2$ when $a=b$.