Question

The integral value of \(\int_0^{\frac{π}{2}} \int_0^{a sinθ} \int_0^r r \,dr \,dθ \,dz \) is _____

(a) 0.5

(b) 0.25

(c) 1

(d) 0

The question was asked during an interview.

This interesting question is from Triple Integral in chapter Multiple Integrals of Engineering Mathematics

Answer 1

The correct option is (d) 0

The explanation is: \(\int_{θ=0}^{\frac{π}{2}} \int_{r=0}^{a sinθ} \int_{z=0}^r r \,dr \,dθ \,dz = \int_{θ=0}^{\frac{π}{2}} \int_{r=0}^{a sinθ} r \big[z\big]_0^r \,dr \,dθ \)

\(= \int_{θ=0}^{\frac{π}{2}} \int_{r=0}^{a sinθ} r^2 \,dr \,dθ\)

\(\int_{θ=0}^{\frac{π}{2}}\big[\frac{r^3}{3}\big]_0^{sin⁡θ} \,dθ = \int_0^{\frac{π}{2}} \frac{sin^3 θ}{3} \,dθ = \int_0^{\frac{π}{2}} \frac{3 sin⁡θ-sin⁡3θ}{12} \,dθ = \Big[\frac{-3 cos⁡θ + 3 cos⁡3θ}{12}\Big]_0^{\frac{π}{2}}=0\).