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

We do this by integrating term-by-term and then applying the limits.

This integral is well known

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The net area between ( f(x) = \csc(x) - \cos(x)\sin(x) ) and the x-axis over ( x ) in ( \left[\frac{\pi}{6}, \frac{5\pi}{8}\right] ) is given by the definite integral ( \int_{\frac{\pi}{6}}^{\frac{5\pi}{8}} |\csc(x) - \cos(x)\sin(x)| , dx ). Since the integrand involves absolute value, we need to consider the intervals where ( \csc(x) - \cos(x)\sin(x) ) is positive and negative separately. This integral can be challenging to compute directly, and you may need to use numerical methods or software to obtain a numerical value.

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