Solutions:

(a) If we rotate the ellipsoid around the x-axis the surface area, S, is twice the area of that in the first quadrant.

Factoring out the from the square root, and letting

Using a table of integrals (#30 in Stewart)

(b)

Set

Using a table of integrals (#21 in Stewart)

Computations with g:

Consequently:

This is the same as the answer in the book:

because

and