Question

Volume of an object expressed in spherical coordinates is given by \(V = \int_0^{\frac{π}{2}}∫_0^{\frac{π}{3}}∫_0^1r cos∅ dr d∅ dθ.\) The value of the integral is _______

(a) \(\frac{\sqrt{3}}{2}\)

(b) \(\frac{1}{√2} π\)

(c) \(\frac{\sqrt{3}}{2} π\)

(d) \(\frac{\sqrt{3}}{8}π\)

This question was posed to me during an online interview.

My enquiry is from Variable Treated as Constant topic in portion Partial Differentiation of Engineering Mathematics

Answer 1

Right choice is (d) \(\frac{\sqrt{3}}{8}π\)

The best I can explain: Given: \(V = \int_0^{\frac{π}{2}}∫_0^{\frac{π}{3}}∫_0^1r cos∅ \,dr \,d∅ \,dθ\)

\(V = \int_0^{\frac{π}{2}}∫_0^{\frac{π}{3}}(\frac{r^2}{2})_0^1cos∅ \,d∅ \,dθ\)

\(V = \frac{1}{2} \int_0^{\frac{π}{2}}(sin∅)_0^{\frac{π}{3}}d∅ \,dθ\)

\(V = \frac{1}{2}×\frac{\sqrt{3}}{2}\int_0^{\frac{π}{2}}dθ\)

\(V = \frac{1}{2}×\frac{\sqrt{3}}{2}×\frac{π}{2}\)

\(V = \frac{\sqrt{3}}{8}π \)