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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 -

    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

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