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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 -

    The triangle is bounded by
    x-axis
    x=1, and
    y=x.

    The integrals can be carried out in order dx, then dy or vice-versa.
    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

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