Finding Volume of Solid Generated by Revolving Cycloid Arch Around x-Axis

[Yuma]

1. Find the volume of the solid generated by revolving the region bounded by the x-axis and one arch of the cycloid x=theta-sin(theta), y=1-cos(theta) around the x-axis.



2. hint:dV=(pi)y^2 dx



3. So far I have been unable to solve for theta so that I can form a relationship between the two equations. I don't believe that solving the x= equation for theta can be done, and the integral that I come up with if I solve the y= equation for theta and substitute it into the x= equation is so unwieldy that I don't believe it is the right one. Besides, the shape is revolving around the x-axis, so I should have a y= equation in order to proceed with solving this problem in the way that I was taught. Any help is much appreciated!

 

[HallsofIvy]

1. Find the volume of the solid generated by revolving the region bounded by the x-axis and one arch of the cycloid x=theta-sin(theta), y=1-cos(theta) around the x-axis.



2. hint:dV=(pi)y^2 dx



3. So far I have been unable to solve for theta so that I can form a relationship between the two equations. I don't believe that solving the x= equation for theta can be done, and the integral that I come up with if I solve the y= equation for theta and substitute it into the x= equation is so unwieldy that I don't believe it is the right one. Besides, the shape is revolving around the x-axis, so I should have a y= equation in order to proceed with solving this problem in the way that I was taught. Any help is much appreciated!

dV= (pi)y^2 dx. y= 1- cos(theta) and, since x= theta- sin(theta), dx= [1- cos(theta)]dtheta.