GR9367, Q 63

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GR9367, Q 63

Postby goingunder » Fri Oct 09, 2009 5:52 am


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.
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))


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Re: GR9367, Q 63

Postby diogenes » Fri Oct 09, 2009 6:15 am

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.

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