Question

What is the volume of the sphere of radius ‘a’?

(a) \(\frac{4}{3} πa \)

(b) 4πa

(c) \(\frac{4}{3} πa^2 \)

(d) \(\frac{4}{3} πa^3 \)

This question was posed to me by my school principal while I was bunking the class.

This key question is from Volume of Solid of Revolution in portion Integral Calculus of Engineering Mathematics

Answer 1

Right option is (d) \(\frac{4}{3} πa^3 \)

To explain I would say: The equation of a circle is x^2 + y^2 = a^2

When it is revolved about x-axis, the volume is given as

\(V = 2 \int_a^b πy^2 dy\)

\(= 2 \int_0^a π(a^2-x^2) dx\)

\(= 2π \Big(a^2 x – \frac{x^3}{3}\Big)_0^a\)

\(= \frac{4}{3} πa^3.\)