Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ .
A) $40.5 \pi$
B) $27 \pi$
C) $81 \pi$
D) $$324 \pi$$
E) $162 \pi$

Question

Quizplus · Accepted Answer

The region of integration is the disk $x^2 + y^2 \leq 9$, which in polar coordinates is $0 \leq r \leq 3$. The volume is then$$\begin{split} V & = \iint_D (x^2 + y^2) dA \\ & = \int_0^{2\pi} \int_0^3 r^2 \ r \ dr \ d\theta \\ & = \int_0^{2\pi} \left[ \frac{r^4}{4} \right]_0^3 \ d\theta \\ & = \int_0^{2\pi} \frac{81}{4} \ d\theta \\ & = \frac{81}{4} \cdot 2\pi \\ & = 40.5 \pi. \end{split}$$

Use Polar Coordinates to Find the Volume of the Solid z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Use Polar Coordinates to Find the Volume of the Solid $z = x ^ { 2 } + y ^ { 2 }$