Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 25$.
A) $2,500 \pi$
B) $625 \pi$
C) $312.5 \pi$
D) $1,250 \pi$
E) $208.3 \pi$

Question

Quizplus · Accepted Answer

The volume under the paraboloid $z = x^2 + y^2$ and above the disk $x^2 + y^2 \leq 25$ can be found using polar coordinates. In polar coordinates, $x = r\cos(\theta)$ and $y = r\sin(\theta)$, so $z = r^2$. The area element in polar coordinates is $dA = r dr d\theta$. The limits for $r$ are from 0 to 5 (the radius of the disk), and for $\theta$ are from 0 to $2\pi$ (a full circle). The volume integral in polar coordinates becomes $\int_{0}^{2\pi} \int_{0}^{5} r^2 r dr d\theta = \int_{0}^{2\pi} \int_{0}^{5} r^3 dr d\theta$. Solving this integral gives $\frac{1}{4}r^4 |_{0}^{5} = \frac{625}{4}$ for the $r$ integral, and multiplying by $2\pi$ for the $\theta$ integral gives $312.5\pi$.

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 $z = x ^ { 2 } + y ^ { 2 }$