Use polar coordinates to find the volume of the solid under the paraboloid and above the disk .

Question

Use polar coordinates to find the volume of the solid under the paraboloid  and above the disk  .

Quizplus · Accepted Answer

The paraboloid can be expressed in polar coordinates as $z=r^2$. The disk is defined by $r\leq \sqrt{3}$ in the $xy$-plane. Therefore, the limits of integration are $0\leq 	heta \leq 2\pi$ and $0\leq r\leq \sqrt{3}$. The volume of the solid is given by the integral:
\begin{align*}
\iiint_V dV &= \int_0^{2\pi} \int_0^{\sqrt{3}} \int_0^{r^2} r\ dz\ dr\ d	heta \
&= \int_0^{2\pi} \int_0^{\sqrt{3}} r^3\ dr\ d	heta \
&= \int_0^{2\pi} \frac{1}{4} (3)^2\ d	heta \
&= \frac{3}{4} (2\pi) \
&= \frac{3\pi}{2}.
\end{align*}
Converting to decimal form, we get the final answer of 11eaae29_f447_31f2_87fe_1f21bdf8edd9_TB2067_11.

Use Polar Coordinates to Find the Volume of the Solid