Hello, acosta0809!

Let $\displaystyle F(a)$ be the function which gives the area under the graph of $\displaystyle y\,=\,xe^{-x}$

between $\displaystyle x=0$ and $\displaystyle x=a$ for $\displaystyle a>0.$

i found $\displaystyle F(a)$ to be: .$\displaystyle -xe^{-x} -e^{-x}$

then evaluated with the given endpoints i got: .$\displaystyle -ae^{-a} -e^{-a}+1$

(not sure if this is correct) It's correct!

The second question is:

Is $\displaystyle F(a)$ an increasing or a decreasing function? Find the first derivative (slope).

We have: .$\displaystyle F(a) \:=\:ae^{-a} - e^{-a} + 1$

Then: .$\displaystyle F'(a) \;=\;ae^{-a} - e^{-} + e^{-a} + 0 \;=\;\frac{a}{e^a} $

Since $\displaystyle a > 0$, then: .$\displaystyle F'(a) \:=\:\frac{a}{e^a}$ is always positive.

Therefore, $\displaystyle F(a)$ is an increasing function.

If we look at the graph, it's obvious . . . Code:

. . . - |
. . . - | ..*.
. . . - | *::::::*.
. . . - | *::::::::::*..
. . . - |*::::::::::::::::*
. . . - |:::::::::::::::::| *
. . . - * - - - - - - - - + - - - - -
. . . - | a

As $\displaystyle a$ increases, the area under the curve increases.