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

May 30, 2017

$\text{Area} \approx - 2.5523$

#### Explanation:

Here is a graph of $f \left(x\right)$:

graph{sqrtx - x [-10, 10, -5, 5]}

Please observe that $f \left(x\right)$ is negative over the interval, therefore, the area will be negative.

$\text{Area} = {\int}_{2}^{4} {x}^{\frac{1}{2}} - x \mathrm{dx}$

$\text{Area} = {\left(\frac{2}{3} {x}^{\frac{3}{2}} - \frac{1}{2} {x}^{2}\right]}_{2}^{4}$

$\text{Area} = \left(\frac{2}{3} {\left(4\right)}^{\frac{3}{2}} - \frac{1}{2} {\left(4\right)}^{2}\right) - \left(\frac{2}{3} {\left(2\right)}^{\frac{3}{2}} - \frac{1}{2} {\left(2\right)}^{2}\right)$

$\text{Area} \approx - 2.5523$