Solve the problem.
-Find the volume of the solid generated by revolving the region enclosed by the ellipse $4 x ^ { 2 } + 25 y ^ { 2 } = 100$ about the $x$-axis.
A) $4 \pi ^ { 2 }$
B) $\frac { 160 \pi } { 3 }$
C) $\frac { 80 \pi } { 3 }$
D) $\frac { 40 \pi } { 3 }$

Question

Quizplus · Accepted Answer

We can rewrite the given equation as $\frac{x^2}{5^2}+\frac{y^2}{2^2}=1$. This implies that the semi-axes of the ellipse are $a=5$ and $b=2$.
The volume of the solid generated by revolving the region bounded by the given ellipse about the $x$-axis is given by
$$V=\pi\int_{-a}^{a}y^2dx.$$
By using the equation of the ellipse, we have
$$y=\frac{\sqrt{100-4x^2}}{5}.$$
Thus,
$$V=\pi\int_{-5}^{5}\left(\frac{\sqrt{100-4x^2}}{5}\right)^2dx=\frac{80\pi}{3}.$$
Therefore, the answer is $\boxed{\textbf{(C)}}$.

Solve the Problem 4x2+25y2=1004 x ^ { 2 } + 25 y ^ { 2 } = 1004x2+25y2=100

Solve the Problem $4 x ^ { 2 } + 25 y ^ { 2 } = 100$