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

##### 1 Answer

Apr 11, 2018

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

Since we're rotating around the x-axis, the radius of the circle 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

Therefore, the surface area of the shape produced by rotating the function

*Final Answer*