Question

Evaluate \(\int_0^∞ \int_x^∞ \frac{e^{-y}}{y} dydx \) by changing the order of integration.

(a) 0

(b) 1

(c) 2

(d) 1/2

This question was addressed to me in a national level competition.

This interesting question is from Surface Integrals in portion Vector Integral Calculus of Engineering Mathematics

Answer 1

Right answer is (b) 1

The explanation is: In the question, y: x to infnity

x: 0 to infinity

Now changing the orrder of integration:

y=x

y tends to infinity

y: 0 to infinity

x: 0 to y

\(\int_0^∞ \int_0^y \frac{e^{-y}}{y} dydx \)

\( = \int_0^∞ \frac{e^{-y}}{y} ydy \)

= -(0-1)

= 1.