Integrate f(x,y) Over the Set A

[JG89]

Homework Statement

Integrate f(x,y) = e^{-(x^2 + y^2)} over the set A = \{ (x,y): x > 0, y > 0, x^2 + y^2 < a \}

Homework Equations

The Attempt at a Solution

The polar coordinate transformation g(r,\theta) = (r cos\theta, r sin\theta) is a diffeomorphism from A to the set B = \{(r,\theta): 0 < r < a, 0 < \theta < \pi / 2 \}, so I can use the change of variables theorem.

So \int_A e^{-(x^2 + y^2)} = \int_B re^{-r^2} = \int_0^a \int_0^{\pi/2} re^{-r^2} d \theta dr = (\pi / 2) \int_0^a re^{-r^2} dr.

Let u = -r^2. Then -du/2 = rdr, so we have (\pi / 2) \int_0^a re^{-r^2} dr = (- \pi / 4) \int_0^{-a^2} e^u du = (- \pi / 4)(e^{-a^2} - 1).

I get the same answer if I reverse the order of integration. Is this answer correct?

 

[SammyS]

It all looks fine to me.

 

[DavidAlan]

All good buddy. Why even mention diffeomorphisms? This be calculus, there be dragons there.

 

[JG89]

Because as far as I know, the change of variables theorem requires g to be a C^r diffeomorphism.