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

Integrate the given function and evaluate using the given limits.

Split the integral:

The right can be solved after a simple substitution.

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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]).

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