What is the Shell Method for Evaluating Volume of Revolved Solids?

[icesalmon]

Homework Statement

Using the shell method, set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.
y = x³ bounded by y = 8, and x = 0.

Homework Equations

volume of a solid revolved around x-axis using the shell method.

The Attempt at a Solution

I have to find the height h(y) of some arbitrary shell and the average radius p(y) of all shells multiply them together and then integrate them, which I am fairly confident I can do so I am not going to worry about that at this time. For my height, h(y), I have a distance of "y" and I understand that. But the average radius p(y) is what I do not understand, the answer says that they are just multiplying the height by x(y), y^1/3, which they do because they are integrating wrt y. But geometrically, I would have thought the average radius would be (8 - y^1/3).

 

[eumyang]

I have to find the height h(y) of some arbitrary shell and the average radius p(y) of all shells multiply them together and then integrate them, which I am fairly confident I can do so I am not going to worry about that at this time. For my height, h(y), I have a distance of "y" and I understand that. But the average radius p(y) is what I do not understand, the answer says that they are just multiplying the height by x(y), y^1/3, which they do because they are integrating wrt y. But geometrically, I would have thought the average radius would be (8 - y^1/3).

Are you working out of the Larson book, by chance?
You got it mixed up, I think. y = 8 is a horizontal line, and it serves as the upper limit of integration. Your representative rectangles are parallel to the axis of revolution. h(y) is the distance from the y-axis to the curve, so it's actually y^1/3. p(y) is the distance from the rectangle to the x-axis, which is y.

 

[icesalmon]

I understand it now, thank you. I am working out of larson 9e, is there another textbook for calculus that you can suggest would help teach the course better?