# 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

#### Explanation:

Integrate the given function and evaluate using the given limits.

#int_1^4(x-sqrt(x+1))dx#

Split the integral:

#int_1^4xdx-int_1^4sqrt(x+1)dx#

The left is a basic integral, yielding

The right can be solved after a simple substitution.

#u=x+1, du=dx#

We will also have to modify our limits of integration for this integral. We have stated that

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

Using that

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

Evaluating, we get