# What is the net area between f(x)=(x-3)e^x in x in[1,2]  and the x-axis?

Feb 21, 2016

$2 {e}^{2} - 3 e$

#### Explanation:

Area should always be positive.

Since $f \left(x\right) < 0$ for all $x \in \left[1 , 2\right]$, the area is given by

${\int}_{1}^{2} | f \left(x\right) | \text{d"x = int_1^2 -f(x) "d} x$

$= {\int}_{1}^{2} \left(3 - x\right) {e}^{x} \text{d} x$

$= {\left[\left(4 - x\right) {e}^{x}\right]}_{1}^{2}$

$= 2 {e}^{2} - 3 e$