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!

## GR9367, Q 63

### 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.

$$x^2+y^2$$, 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.