Question

Evaluate \(\int_0^∞ \int_0^{π/2} e^{-r^{2}} rdθdr \).

(a) \( \pi \)

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

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

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

The question was asked in unit test.

Enquiry is from Surface Integrals in division Vector Integral Calculus of Engineering Mathematics

Answer 1

Right choice is (c) \( \frac{\pi}{4} \)

To explain I would say: The integral is in polar coordinates.

Substitute r^2 as t

\(\int_0^∞ \int_0^{π/2} e^{-t} dθ \frac{dt}{2} \)

\(= \frac{1}{2} \int_0^{π/2}Γ(1)dθ \)

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