Problems

01

Volumes of solids

For each of the following regions, write and compute a double integral to find its volume.

• (a) Region bounded by $y=0$, $y=x$, $z=0$, $z=x$, $x=0$, $x=5$.
• (b) Region bounded by $x^2+y^2+z^2=r^2$. (You will need to add two double integrals.)
• (c) Region bounded by $x=0$, $x=2$, $y=0$, $y=+\infty$, $z=\frac{1}{1+y^2}$.
• (d) Region under $z=4-(x^2+2y^2)$ and above $z=0$. (Hint: use a substitution $x=2\sin (\theta )$ and the Wallis Formula: ${\int }_0^{\pi /2}{\cos }^4(\theta )d\theta =3\pi /16$.)