calculus
posted by mabel .
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)

The triangle is bounded by
xaxis
x=1, and
y=x.
The integrals can be carried out in order dx, then dy or viceversa.
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