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
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
1 answer