Question

What is the volume generated when the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is revolved about its minor axis?

(a) 4 ab cubic units

(b) \(\frac{4}{3} a^2 b \) cubic units

(c) \(\frac{4}{3} ab \) cubic units

(d) 4 cubic units

I have been asked this question in unit test.

Question is taken from Volume of Solid of Revolution topic in section Integral Calculus of Engineering Mathematics

Answer 1

Right option is (b) \(\frac{4}{3} a^2 b \) cubic units

To explain I would say: y- axis is the minor axis. \(x^2 = \frac{a^2}{b^2} (b^2 – y^2)\)

\(V = \int_a^b πx^2 dy\)

\(= \int_{-b}^b π \frac{a^2}{b^2} (b^2 – y^2) \,dy \)

\(= 2π \frac{a^2}{b^2}  \Big(b^2 y- \frac{y^3}{3}\Big)_0^b \)

\(= 2π \frac{a^2}{b^2}  (b^3- \frac{b^3}{3}) \)

\(= \frac{4}{3} a^2 b \) cubic units.