Volume of a cone expressed as a solid of revolution?

[rapid dagger]

How would I do this problem? Do I just integrate with circular cross-sections?

[inner wedge]

I mean, cylindrical cross sections, technically.

[rapid dagger]

I thought cross sections were two-dimensional?

[inner wedge]

They're three dimensional. It doesn't make sense to take the volume of a two dimensional object. It's just that we mark one of the dimensions dx and take the limit as it goes to 0.

The dx dimension is height for washer method, thickness for shell method.

[rapid dagger]

so the dx is included in the cross section?

I thought the cross section just meant the 2-d face you'd get if you cut a 3-d object

[inner wedge]

Again, no, because again, the volume of a 2D object is 0.

Look, you know Riemann sums?

[rapid dagger]

yes

[inner wedge]

It's like that. It doesn't make sense to take the area of a line, so you take the area of a rectangle and let its width tend to zero. Same deal.

In fact, it's literally exactly the same deal. The washers and cylindrical shells are just the rotations of the Riemann rectangles.

[rapid dagger]

thanks