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calculus

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find the average value of the function f(x,y)=e^(-x^2) over the plane region R which is the triangle with vertices (0,0), (1,0) and (1,1)

  • calculus -

    Please check the function for typo, since
    f(x,y)=e^(-x²) is independent of y.

    Assuming no typo,
    the region R is bounded between x=0 and x=1, and y=0 and y=x.

    So the integegration
    dy from 0 to x
    dx from 0 to 1.

    I=∫∫ye^(-x²)dy dx
    =∫xe^-x² dx
    =1/2-e^(-1)/2

    The area of R is 1/2, so
    Average value
    = I/(1/2)
    =(e-1)/e

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