# What is the net area between f(x) = x-sqrt(x+1)  and the x-axis over x in [1, 4 ]?

##### 1 Answer
Jul 15, 2017

$\frac{15}{2} - \frac{2}{3} \left({5}^{\frac{3}{2}} - {2}^{\frac{3}{2}}\right)$

#### Explanation:

Integrate the given function and evaluate using the given limits.

${\int}_{1}^{4} \left(x - \sqrt{x + 1}\right) \mathrm{dx}$

Split the integral:

${\int}_{1}^{4} x \mathrm{dx} - {\int}_{1}^{4} \sqrt{x + 1} \mathrm{dx}$

The left is a basic integral, yielding 1/2x^2]_1^4

The right can be solved after a simple substitution.

$u = x + 1 , \mathrm{du} = \mathrm{dx}$

We will also have to modify our limits of integration for this integral. We have stated that $u = x + 1$, so we now have $u \in \left[2 , 5\right]$

1/2x^2]_1^4-int_2^5sqrt(u)du

Using that $\sqrt{u} = {u}^{\frac{1}{2}}$, we have:

1/2x^2]_1^4-2/3u^(3/2)]_2^5

Evaluating, we get $\frac{15}{2} - \frac{2}{3} \left({5}^{\frac{3}{2}} - {2}^{\frac{3}{2}}\right)$