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

Question

Emma Aldridge · Accepted Answer

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

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))#

Emilia Aguilar · Answer

undefined To find the net area between $ f(x) = x - \sqrt{x + 1} $ and the x-axis over $ x $ in $[1, 4]$, you need to integrate $ f(x) $ with respect to $ x $ over the given interval and take the absolute value of the result.

First, find the points of intersection between $ f(x) $ and the x-axis by solving $ f(x) = 0 $. Set $ x - \sqrt{x + 1} = 0 $ and solve for $ x $. This gives $ x = 1 $ as the only solution in the interval $[1, 4]$.

Next, integrate $ f(x) $ from $ x = 1 $ to $ x = 4 $ and take the absolute value of the result:

$$
\text{Net area} = \left| \int_{1}^{4} (x - \sqrt{x + 1}) \, dx \right|
$$

Integrate $ x - \sqrt{x + 1} $ with respect to $ x $ from 1 to 4 using appropriate integration techniques or software.

After integration, take the absolute value of the result to obtain the net area between $ f(x) $ and the x-axis over $ x $ in $[1, 4]$.