Chapter 14.8, Problem 34E

VolumeUse the result of Exercise 33 to find the volume of each dome-shaped solid lying below the surface $z=f(x,y)$ and above the elliptical region $R$. (Hint: After making the change of variables given by the results in Exercises 33, make a second change of variable to polar coordinates.)

(a) x $f(x,y)=16-x^2-y^2;R:\frac{x^2}{16}+\frac{y^2}{9}\le 1$

(b) $f(x,y)=A\cos \left(\frac{\pi }{2}\sqrt{\frac{x^2}{a^2}+\frac{y^2}{b^2}}\right);R:\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1$