Question

Evaluate \(\int_{-1}^1 \int_0^z \int_{x-z}^{x+z}(x+y+z)dxdydz.\)

(a) 0

(b) 1

(c) 2

(d) 3

This question was posed to me in an online interview.

My enquiry is from Volume Integrals in division Vector Integral Calculus of Engineering Mathematics

Answer 1

Right option is (a) 0

The best explanation: \(\int_{-1}^1 \int_0^z \int_{x-z}^{x+z}(x+y+z)dxdydz=

\int_{-1}^1 \int_0^z (xy+\frac{y^2}{2}+zy)dxdz \) from x-z to x+z

\(= \int_{-1}^1\int_0^z(3x^2+z^2+2xz)dxdz= \int_{-1}^1(x^3+z^2 x+x^2 z)dz \) from 0 to z

\(= \int_{-1}^1(z^3+z^3+z^3)dz= \frac{z^4}{4} \) from -1 to 1

= 0.