What is the surface area produced by rotating f(x)=cscxcotx, x in [pi/8,pi/4] around the x-axis?

Apr 11, 2018

To find the surface area of the figure, we need to integrate the circumference of the figure with respect to $x$.

Since we're rotating around the x-axis, the radius of the circle is $f \left(x\right)$. Therefore, the surface area is:

${\int}_{\frac{\pi}{8}}^{\frac{\pi}{4}} \left(2 \pi r\right) \mathrm{dx}$

$2 \pi {\int}_{\frac{\pi}{8}}^{\frac{\pi}{4}} \csc x \cot x \mathrm{dx}$

2pi (-cscx)]_(pi/8)^(pi/4)

$- 2 \pi \left(\csc \left(\frac{\pi}{4}\right) - \csc \left(\frac{\pi}{8}\right)\right)$

$- 2 \pi \left(\sqrt{2} - \sqrt{\frac{2}{1 - \cos \left(\frac{\pi}{4}\right)}}\right) \text{ "" }$ (using half-angle formula)

$- 2 \pi \left(\sqrt{2} - \sqrt{\frac{2}{1 - \frac{\sqrt{2}}{2}}}\right)$

$- 2 \pi \left(\sqrt{2} - \frac{2}{\sqrt{2 - \sqrt{2}}}\right)$

This is the final value, which can be simplified in various ways, as necessary. As a decimal, it is $7.533$.

Therefore, the surface area of the shape produced by rotating the function $f \left(x\right) = \csc x \cdot \cot x$ from $x = \frac{\pi}{8}$ to $x = \frac{\pi}{4}$ is $7.533$.