Question

evaluate the integral integration 0 to infinity, integration 0 to infinity e^-{x²+y²}dydx.by changing to polar coordinates

Answer 1

Answer:

This is an old favorite of mine.

Define

I=∫

+∞

−∞

e−x2dx

Then

I2=(∫

+∞

−∞

e−x2dx)(∫

+∞

−∞

e−y2dy)

I2=∫

+∞

−∞

∫

+∞

−∞

e−(x2+y2)dxdy

Now change to polar coordinates

I2=∫

+2π

0

∫

+∞

0

e−r2rdrdθ

The θ integral just gives 2π, while the r integral succumbs to the substitution u=r2

I2=2π∫

+∞

0

e−udu/2=π