Solve the problem.
-Find the volume of the solid bounded by the ellipsoid $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$ and the planes $z = - 2$ and $z = 2$. (The area of an ellipse with semiaxes a and $b$ is $\pi a b$.)
A) $13,632.00 \pi$ units $^ { 3 }$
B) $176.00 \pi$ units $^ { 3 }$
C) $227.20 \pi$ units $^ { 3 }$
D) $252.80 \pi$ units $^ { 3 }$

Question

Quizplus · Accepted Answer

The volume of the solid is calculated by integrating the area of the ellipse cross-sections from $z = -2$ to $z = 2$. The semi-major and semi-minor axes of the ellipse at a given $z$ are $\sqrt{36(1 - \frac{z^2}{25})}$ and $\sqrt{100(1 - \frac{z^2}{25})}$. The area of the ellipse is $\pi \times \sqrt{36(1 - \frac{z^2}{25})} \times \sqrt{100(1 - \frac{z^2}{25})}$. Integrating this area from $z = -2$ to $z = 2$ gives the volume $227.20 \pi$ units$^3$.

Solve the Problem x236+y2100+z225=1\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 136x2​+100y2​+25z2​=1

Solve the Problem $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$