Question

An ellipse whose major axis is $2a$ and the minor axis is $2b$ revolves about the minor axis. The volume of the solid is equal to

• A. $\frac{4}{3}\pi a{b}^2$
• B. $\frac{2}{3}\pi a{b}^2$
• C. $\frac{2}{3}\pi a^{2}b$
• D. $\frac{4}{3}\pi a^{2}b$

Answer 1

The correct option is D $\frac{4}{3}\pi a^{2}b$

Consider a circle.

We know that it is a special case of an ellipse where length of major axis is equal to the length of minor axis which is equal to the radius of the circle.

Hence its area is $\pi r^2$.

On the contrary, the area occupied by an ellipse is $\pi \left(a\right)\left(b\right)$.

Now revolving a square about its radius gives us a sphere.

The volume of a sphere is $\frac{4\pi r^3}{3}$

Revolving an ellipse creates an ellipsoid which will therefore have a volume similar to a sphere, the volume being

$=\frac{4\pi a^{2}b}{3}$ or $\frac{4\pi a{b}^2}{3}$ , depending upon the axis of rotation.

Hence we have two possibilities.

Case I

The ellipse is rotated about the major axis.

In that case the ellipsoid will be narrow, and hence the volume occupied will be lesser.

Let 2b is the length of the minor axis, and 2a be the major axis.

Since the volume occupied will be lesser, the volume will be

$=\frac{4\pi }{3}{b}^2.a$

Case II

The ellipse is rotated about the minor axis.

In that case the ellipsoid will be broader, and hence the volume occupied will be larger.

Let 2b is the length of the minor axis, and 2a be the major axis.

Since the volume occupied will be larger, the volume will be

$=\frac{4\pi }{3}{a}^2.b$

In the above equation, the axis of rotation is the minor axis, whose length is 2b.

Therefore the volume occupied by the sphere will be $\frac{4\pi }{3}{a}^{2}b$.