Question

Find the first quandrant area under the curve y=xe^-x.

Answer 1

Let us find the first quandrant area under the curve $y=xe^{-x}$. For this firstly let us sketch the graph of $y=xe^{-x}:$

It follows that the area $A$ is

$A=\int_0^{+\infty}xe^{-x}dx=\lim\limits_{a\to +\infty}\int_0^{a}xe^{-x}dx=$

$|u=x, dv=e^{-x}dx,du=dx, v=-e^{-x}|$

$=\lim\limits_{a\to +\infty}(-xe^{-x}|_0^a+\int_0^{a}e^{-x}dx) =\lim\limits_{a\to +\infty}(-ae^{-a}-e^{-x}|_0^a)= \lim\limits_{a\to +\infty}(-ae^{-a}-e^{-a}+1)= \lim\limits_{a\to +\infty}\frac{-a-1}{e^a}+1=$

| let us use the L'Hôpital's Rule |

$=\lim\limits_{a\to +\infty}\frac{-1}{e^a}+1=1$