# calculus

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

• calculus -

The triangle is bounded by
x-axis
x=1, and
y=x.

The integrals can be carried out in order dx, then dy or vice-versa.
However, integrating with respect to dy first makes for an easier integration (in the evaluation of I).

Be very sure you understand how the limits are obtained. Draw a sketch of the triangle, and follow the limits, and it will be easy to visualize how the limits can be found.

Area, A = ∫∫dxdy
Limits of integration
y from 0 to x
x from 0 to 1

Integral of values
I=∫∫xe^ydydx
within the same limits

Average value
= I/A

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