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

The end points for the curved boundary are

The numerical area ( suppressing negative sign )

#=-[x^2/2-1/6x(4x+1)^(3/2)+1/6int(4x+1)^(3/2) dx], for x = 1 to 4

between x = 1 and 4

graph{x(1-sqrt(4x+1)) [0, 50, -25, 0]}

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To find the net area between ( f(x) = x - x\sqrt{4x + 1} ) and the x-axis over ( x ) in the interval ([1, 4]), you first need to integrate the function within the given interval, and then take the absolute value of the result since the function dips below the x-axis. The integral will be:

[ \int_{1}^{4} \left| x - x\sqrt{4x + 1} \right| , dx ]

You can break this integral into two parts since the function changes its behavior at ( x = \frac{1}{4} ). Therefore, the integral becomes:

[ \int_{1}^{4} (x - x\sqrt{4x + 1}) , dx + \int_{\frac{1}{4}}^{4} (x\sqrt{4x + 1} - x) , dx ]

After integrating each part, you should get the net area between the function and the x-axis over ( x ) in the interval ([1, 4]).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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