How would I find the area between these two functions?
y=4xe^-x^2 and y=|x|.
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.