Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 4$. Select the correct answer.
A) $64 \pi$
B) $5.3 \pi$
C) $32 \pi$
D) $16 \pi$
E) $8 \pi$

Question

Quizplus · Accepted Answer

The volume under the paraboloid and above the disk can be found using polar coordinates. The paraboloid equation $z = x^2 + y^2$ becomes $z = r^2$ in polar coordinates. The disk $x^2 + y^2 \leq 4$ translates to $0 \leq r \leq 2$. The volume integral is $\int_{0}^{2\pi} \int_{0}^{2} r^2 \cdot r \, dr \, d\theta$, which evaluates to $8\pi$.

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