# What is the net area between f(x) = x-4  and the x-axis over x in [2, 4 ]?

Dec 15, 2017

See below.

#### Explanation:

Plugging in $x = 2$ and $x = 4$

$2 - 4 = - 2$ . $4 - 4 = 0$

shows us that the area we seek is under the x axis, and so will be negative.

We need to integrate $x - 4$, with upper and lower bounds of $4$ and $2$.

$A = - {\int}_{2}^{4} \left(x - 4\right) \mathrm{dx} = {\left[\frac{1}{2} {x}^{2} - 4 x\right]}_{2}^{4}$

$A = - \left\{{\left[\frac{1}{2} {x}^{2} - 4 x\right]}^{4} - {\left[\frac{1}{2} {x}^{2} - 4 x\right]}_{2}\right\}$

Plugging in upper and lower bounds:

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

$A = - \left\{\left[- 8\right] - \left[- 6\right]\right\} = 2$

$A r e a = 2$ square units

Graph: