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Math Help - Area Between Two Curves

  1. #1
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    Area Between Two Curves

    How would I find the area between these two functions?

    y=4xe^-x^2 and y=|x|.
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  2. #2
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    first find the intersection points ...

    for x \geq 0 ...

    4xe^{-x^2} = x

    4xe^{-x^2} - x = 0

    x(4e^{-x^2} - 1) = 0

    x = 0, and ...

    e^{-x^2} = \frac{1}{4}

    -x^2 = \ln\left(\frac{1}{4}\right)

    x^2 = \ln{4}

    x = \sqrt{\ln{4}}

    for x < 0 ...

    4xe^{-x^2} = -x

    4xe^{-x^2} + x = 0

    x(4e^{-x^2} + 1) = 0

    x = 0 only.


    A = \int_0^{\sqrt{\ln{4}}} 4xe^{-x^2} - x \, dx

    A = \left[-2e^{-x^2} - \frac{x^2}{2}\right]_0^{\sqrt{\ln{4}}}

    you can evaluate the area value yourself.
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