Question

Calculus: sketch and then calculate the area bounded by the graphs y=xe^-x and y=(1/e)x

Answer 1

Here is the sketch:

http://www.wolframalpha.com/input/?i=g%28x%29%3Dxe^-x%3B+f%28x%29%3D%281%2Fe%29x

Now, you must calculate the area. On the graph, we can see the x-bounds, but if you want to calculate, you must set the two graphs equal to each other to find the x-intercepts.

\[{x \over e} = xe^{-x} \]

we can clearly see x=zero would solve this, as well as x=1.

from the graph we can see xe^-x is bigger, so our integral becomes:

\[\int\limits_{0}^{1} (xe^{-x} - {x \over e}) dx \]

\[\int\limits_{0}^{1} xe^{-x} dx - \int\limits_{0}^{1} {x \over e} dx\]

Can you take it from there?

Answer 2

ok does this look right? -e^(-x)(x+1) - (x^2/2e)

Answer 3

and the area i got is 0.080301 does that sounds right? thanks for your help