# What is the net area between #f(x) = sinx - cos^2x# and the x-axis over #x in [0, 3pi ]#?

The net area is

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To find the net area between ( f(x) = \sin(x) - \cos^2(x) ) and the x-axis over ( x ) in the interval ( [0, 3\pi] ), we need to evaluate the definite integral of ( f(x) ) from ( x = 0 ) to ( x = 3\pi ), and take the absolute value of the result since the function dips below the x-axis.

[ A = \int_{0}^{3\pi} |\sin(x) - \cos^2(x)| , dx ]

First, we'll find the critical points of ( f(x) ) by setting ( f(x) = 0 ).

[ \sin(x) - \cos^2(x) = 0 ] [ \sin(x) - (1 - \sin^2(x)) = 0 ] [ \sin(x) - 1 + \sin^2(x) = 0 ] [ \sin^2(x) + \sin(x) - 1 = 0 ]

Using a quadratic formula, let ( u = \sin(x) ): [ u^2 + u - 1 = 0 ]

The solutions to this quadratic equation are ( u = \frac{-1 \pm \sqrt{5}}{2} ). Since ( \sin(x) ) is between -1 and 1 for all real ( x ), the only solution in the range is ( u = \frac{-1 + \sqrt{5}}{2} ).

Therefore, ( \sin(x) = \frac{-1 + \sqrt{5}}{2} ) is the only critical point.

Now, we'll evaluate the integral:

[ A = \int_{0}^{3\pi} |\sin(x) - \cos^2(x)| , dx = \int_{0}^{3\pi} |\sin(x) - (1 - \sin^2(x))| , dx ]

[ = \int_{0}^{3\pi} |\sin^2(x) + \sin(x) - 1| , dx ]

Since ( \sin^2(x) + \sin(x) - 1 = 0 ) at ( x = \frac{-1 + \sqrt{5}}{2} ), we can split the integral into two parts:

[ A = \int_{0}^{\frac{-1 + \sqrt{5}}{2}} (\sin^2(x) + \sin(x) - 1) , dx + \int_{\frac{-1 + \sqrt{5}}{2}}^{3\pi} (\sin(x) - \sin^2(x) - 1) , dx ]

Now, integrate each part separately and take their absolute values. Finally, sum the absolute values of the two integrals to find the net area.

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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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