Question

If ∭R xyz dx dy dz is solved using cylindrical coordinate where R is the region bounded by the planes x=0, y=0,   z=0,  z=1 & x^2+y^2=1 then what is the value of that integral?

(a) 1/24

(b) 1/16

(c) 1/4

(d) 1/2

This question was posed to me at a job interview.

The query is from Change of Variables In a Triple Integral in section Multiple Integrals of Engineering Mathematics

Answer 1

The correct answer is (b) 1/16

To elaborate: x^2+y^2=1→ρ varies from 0 to 1 substituting x=ρ cos ∅, y=ρ sin ∅, z=z

z varies from 0 to1, x=0, y=0→∅ varies from 0 to π/2

thus the given integral is changed to cylindrical polar given by

 \(\int_0^{\frac{π}{2}} \int_ 0^1 \int_0^1 cos⁡∅sin⁡∅ ρ^3 z  \,dz \,dρ \,d∅ = \int_0^{\frac{π}{2}} \int_ 0^1 cos⁡∅sin⁡∅ ρ^3  \Big[\frac{z^2}{2}\Big]_0^1 \,dρ \,d∅\)

 \(\int_0^{\frac{π}{2}} cos⁡∅sin⁡∅ \Big[\frac{ρ^3}{8}\Big]_0^1 \,d∅ = \int_0^{\frac{π}{2}} cos⁡∅sin⁡∅ \frac{1}{8} \,d∅ \)

 put sin ∅=t, dt=cos ∅

t varies from 0 to 1 \(\int_ 0^1 \frac{1}{8} t \,dt = \Big[\frac{t^2}{16}\Big]_0^1 = \frac{1}{16}.\)