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)

1 answer

average value is volume/base area

one boundary of the region is the line y=x

v = ∫[0,1]∫[0,x] xe^y dy dx
= ∫[0,1] (xe^y)[0,x] dx
= ∫[0,1] x(e^x-1) dx
= (e^x(x-1)-x^2/2)[0,1]
= 1/2

area of base = 1/2

avg value = 1