Question

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

Answer 1

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(x)`. Therefore, the surface area is:

`int_(pi/8)^(pi/4)(2pir) dx`

`2piint_(pi/8)^(pi/4) cscxcotxdx`

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

`-2pi(csc(pi/4)-csc(pi/8))`

`-2pi(sqrt(2) - sqrt(2/(1-cos(pi/4))))" "" "` (using half-angle formula)

`-2pi(sqrt(2) - sqrt(2/(1-sqrt2/2)))`

`-2pi(sqrt(2) - 2/sqrt(2-sqrt2))`

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(x) = cscx*cotx` from `x = pi/8` to `x = pi/4` is `7.533`.

Final Answer