# What is the net area between #f(x) = cscx -xsinx# and the x-axis over #x in [pi/6, (5pi)/8 ]#?

The area under

Apply the Sum Rule:

Let

Let's integrate

Let

and

Now let integrate

From what we derived in (3) and letting:

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To find the net area between the curve (f(x) = \csc(x) - x \sin(x)) and the x-axis over the interval (x \in [\frac{\pi}{6}, \frac{5\pi}{8}]), we need to calculate the definite integral of (f(x)) with respect to (x) over that interval and take the absolute value of the result. The net area is the absolute value of the integral because the function crosses the x-axis multiple times within the given interval, resulting in both positive and negative areas that cancel each other out when summed.

The integral can be calculated as follows:

[ \text{Net area} = \left| \int_{\frac{\pi}{6}}^{\frac{5\pi}{8}} (\csc(x) - x \sin(x)) , dx \right| ]

After evaluating this integral, you'll get the net area between the curve and the x-axis over the given interval.

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