Question

Evaluate \(\int_0^∞ \int_0^∞ e^{-(x^2+y^2 )} dxdy \) by changing into polar coordinates.

(a) \( \pi \)

(b) \( \frac{\pi}{2} \)

(c) \( \frac{\pi}{4} \)

(d) \( \frac{\pi}{8} \)

This question was addressed to me in an online interview.

My question is based upon Surface Integrals topic in section Vector Integral Calculus of Engineering Mathematics

Answer 1

Correct answer is (c) \( \frac{\pi}{4} \)

For explanation I would say: \( \int_0^∞ \int_0^∞ e^{-(x^2+y^2 )} dxdy \)

\(= \int_0^{π/2} \int_0^∞ e^{-(r^2 )} drdθ \)

Substitute \(r^2\) as t

\(= \frac{1}{2} \int_0^{π/2} \int_0^∞ e^{-t} dtdθ \)

\( = \frac{1}{2} \int_0^{π/2}Γ(1)dθ \)

\( = \frac{\pi}{4}. \)