B What Methods Calculate the Volume of a Solid Revolved Around Non-Standard Axes?

[EverythingIsACircle]

Find the volume of the solid generated by revolving the region bounded by y = 4x-x^2 and y = 2x about the y-axis. About the line x = 2.

Since this is rotated about the y-axis, I know I have to manipulate the equation so I can get to x = something. The problem is I cannot change the equation y = 4x-x^2.

Questions:
How would you find the volume that is rotated about the y-axis while the equations are y = form instead of x =? If it is not possible, then what do I have to do in this case?
How would you find the volume if the axis of revolution is not y = 0?

 

[andrewkirk]

You need to express x in terms of y. Use the completing the square technique.

How would you find the volume if the axis of revolution is not y = 0?

Change variables. Replace $x$ by $x'=x-2$.

 

[HallsofIvy]

You can use the "cylinder method" rather that the "disk method" to integrate with respect to x. Imagine a line extending upward from each value of x. That line cuts the "region bounded by y = 4x-x^2 and y = 2x", for x between 0 and 2, in a line segment of length (4x- x^2)- 2x= 2x-x^2. Rotating around the y-axis that forms a cylinder of radius x and height 2x- x^2 so area 2\pi x(2x- x^2)= 2\pi (2x^2- x^3). Taking the "thickness" of each cylinder to be "dx", the whole volume is given by 2\pi \int_0^2 2x^2- x^3 dx.