Solid of Revolution Issue

Question

I need to create a solid of revolution for an ice cream cone. Will these 2 equations do it or will the gap between them cause a problem? Thanks.

$y=\frac{x}{2.5}\left\{0\le x<6\right\}$

$y=\sqrt{-\left(x-6\right)^2+4}\left\{6\le x<8\right\}$

Answer 1

Even though it's a disconnected curve, that won't affect the volume for the solid of revolution. This is because, with respect to the volume, the discontinuity is of "measure zero". That means, in this case, "an ultra-thin piece of paper has no volume." In other contexts, it means "A point has no length" or "a line has no area".

Note that this does not work with the surface area. Even if you set up the integral for the two curves to find surface of revolution, you would be missing the washer-like area between the cone and the ice cream, with outer radius $2.4$ and inner radius $2$.