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Thread: 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 $\displaystyle x \geq 0$ ...

    $\displaystyle 4xe^{-x^2} = x$

    $\displaystyle 4xe^{-x^2} - x = 0$

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

    $\displaystyle x = 0$, and ...

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

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

    $\displaystyle x^2 = \ln{4}$

    $\displaystyle x = \sqrt{\ln{4}}$

    for $\displaystyle x < 0$ ...

    $\displaystyle 4xe^{-x^2} = -x$

    $\displaystyle 4xe^{-x^2} + x = 0$

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

    $\displaystyle x = 0$ only.


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

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

    you can evaluate the area value yourself.
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