Find the volume of the indicated region.
-the region bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane $z = 49$
A) $\frac { 2401 } { 2 } \pi$
B) $\frac { 2401 } { 4 } \pi$
C) $\frac { 2401 } { 6 } \pi$
D) $\frac { 2401 } { 3 } \pi$

Question

Quizplus · Accepted Answer

The volume of the region can be found using the method of cylindrical coordinates. The paraboloid $z = x^2 + y^2$ intersects the plane $z = 49$ in a circle of radius $\sqrt{49} = 7$. The volume can be calculated by integrating the area of circular slices from $z = 0$ to $z = 49$. The integral is $\int_0^{49} \pi r^2 dz$, where $r^2 = z$, giving $\int_0^{49} \pi z dz = \frac{1}{2} \pi z^2 \Big|_0^{49} = \frac{2401}{2} \pi$.

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

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