Chapter 11, Problem 7PS

Volume

(a) Find the volume of the solid bounded below by the paraboloid

$z=x^2+y^2$

and above by the plane $z=1.$

(b) Find the volume of the solid bounded below by the elliptic paraboloid

$z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$

and above by the plane $z=k$, where $k>0.$

(c) Show that the volume of the solid in part (b) is equal to one-half the product of the area of the base times the altitude, as shown in the figure.