## GR9367, Q 63

Forum for the GRE subject test in mathematics.
goingunder
Posts: 6
Joined: Mon Sep 07, 2009 11:23 am

### GR9367, Q 63

Hi,

Any help with GR9367 question 63?

Let R be the circular region of the xy plane with the center at the origin and radius 2.
Then:
Double Integral (over R) of e ^ -(x^2 + y^2) dx dy = ?

A. 4 * Pi
B. Pi *exp(-4)
C. 4* Pi * exp(-4)
D. Pi * (1 - exp(-4))
E. 4*Pi*(exp(1) - exp(-4))

Thanks!

diogenes
Posts: 73
Joined: Sun Aug 31, 2008 9:31 pm

### Re: GR9367, Q 63

The key here seems to be the power of the exponential (along with the fact we have a nice smooth curve to integrate over):

$$x^2+y^2$$, which implies we can convert this problem to polar coordinates, with $$r^2 = x^2+y^2$$ and
$$\theta \ge 0$$ and $$\theta \le 2\pi$$. So, then you can use a u substitution and integrate the following integral:

$$\int_0^{2\pi}\int_0^2 re^{-r^2} \, dr d\theta$$

from there to find that D is the correct solution.