Question

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

(a) \(\frac{√3}{2}\)

(b) \(\frac{1}{√2} π\)

(c) \(\frac{√3}{2}π\)

(d) \(\frac{√3}{4} π\)

This question was addressed to me in an interview.

This is a very interesting question from Change of Order of Integration: Double Integral in chapter Multiple Integrals of Engineering Mathematics

Answer 1

The correct answer is (d) \(\frac{√3}{4} π\)

For explanation: Given: \(V = ∫_0^2π∫_0^\frac{π}{3}∫_0^1 r cos∅ \,dr \,d∅ \,dθ.\)

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

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

\(V = \frac{1}{2}×\frac{√3}{2} ∫_0^2π dθ\)

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

\(       V = \frac{√3}{2} π \)