# What is the volume of the solid produced by revolving #f(x)=cscx-cotx, x in [pi/8,pi/3] #around the x-axis?

A pretty ugly answer, but I got:

and Wolfram Alpha agrees!

Unfortunately, it's the simplest exact numerical solution, apparently, we can't make it look any nicer. :)

DISCLAIMER: The integral is not that hard, but this answer requires a lot of simplification work!

First, let's see how this graph looks.

graph{(cscx - cotx) [0.3927, 1.047, 0, 0.8]}

Here we can see it's a simple curve within this interval. Along the x-axis, the easiest way to do this is to form discs that are perpendicular to the x-axis.

The integral in general is:

If you recall...

Now we're ready to evaluate each one.

Use common denominators to merge some fractions, distributing negative signs over parentheses carefully.

Here we multiply by a unit fraction to get rid of the outer radical in the denominator.

Cross-multiply to merge fractions again.

Expand, and then cancel out anything you can.

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To find the volume of the solid produced by revolving the function ( f(x) = \csc x - \cot x ) on the interval ( \left[\frac{\pi}{8}, \frac{\pi}{3}\right] ) around the x-axis, you can use the method of cylindrical shells.

The volume ( V ) is given by the integral:

[ V = \int_{\frac{\pi}{8}}^{\frac{\pi}{3}} 2\pi x \left(\csc x - \cot x\right) , dx ]

Evaluate this integral to find the volume of the solid.

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