Posted by **soffy** on Tuesday, June 12, 2012 at 9:01am.

find the average value of the function f(x,y)=e^(-x^2) over the plane region R which is the triangle with vertices (0,0), (1,0) and (1,1)

- calculus -
**MathMate**, Tuesday, June 12, 2012 at 10:30am
Please check the function for typo, since

f(x,y)=e^(-x²) is independent of y.

Assuming no typo,

the region R is bounded between x=0 and x=1, and y=0 and y=x.

So the integegration

dy from 0 to x

dx from 0 to 1.

I=∫∫ye^(-x²)dy dx

=∫xe^-x² dx

=1/2-e^(-1)/2

The area of R is 1/2, so

Average value

= I/(1/2)

=(e-1)/e

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