Question

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

Answer 1

`(15)/2-2/3(5^(3/2)-2^(3/2))`

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 `1/2x^2]_1^4`

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 `u=x+1`, so we now have `u in [2,5]`

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

Using that `sqrtu=u^(1/2)`, we have:

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

Evaluating, we get `(15)/2-2/3(5^(3/2)-2^(3/2))`