Find the volume of the indicated region.
-the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 25$
A) $\frac { 2500 } { 3 } \pi$
B) $\frac { 625 } { 2 } \pi$
C) $\frac { 1875 } { 2 } \pi$
D) $\frac { 625 } { 3 } \pi$

Question

Quizplus · Accepted Answer

We can use cylindrical coordinates to find the volume.
The equation of the paraboloid can be written as $z=r^2$, and the equation of the cylinder can be written as $r=5$. 
Limits of integration: 
$$0\leq r\leq 5$$
$$0\leq	heta\leq2\pi$$
$$0\leq z\leq r^2$$
The integral to find the volume is:
$$\int_{0}^{2\pi}\int_{0}^{5}\int_{0}^{r^2}r\,dz\,dr\,d	heta$$
Solving the integral, we get:
$$\frac{625}{2}\pi$$
Therefore, the answer is B.

Find the Volume of the Indicated Region z=x2+y2z = x ^ { 2 } + y ^ { 2 }z=x2+y2

Find the Volume of the Indicated Region $z = x ^ { 2 } + y ^ { 2 }$