Calculating Volume of Object with Annular Slicing Method

[JordieW]

Homework Statement
The question asks; An object is made of a hemisphere of radius 'R' with a hole of radius 'a' drilled through its center of symmetry, as shown in the figure. Use annular slicing to find the volume of the object.

The attempt at a solution
I have managed to calculate the volume using the horizontal slicing method to be;

$\frac{2}{3} π(R^{2}-a^{2})^{\frac{3}{2}}$

Using;

$V = \int_{body} dv(y)$ where $dv(y) = π(R^{2}-y^{2}-a^{2})dy$ where y is height.

I can get the volume of the whole sphere using the annular slicing method but am unsure about how to subtract the volume of the hole from this and cannot find any information on how it is done using this method. Any help would be greatly appreciated!

 

[Dick]

I don't see any difference between "annular slicing" and "horizontal slicing". I think you have already done the problem correctly. You could use slicing on the hole to figure out what to remove from the hemisphere, but I don't think that makes it any easier or different.

 

[JordieW]

Thanks for the quick reply. The lecture notes I have been given make it seem as there is a difference between the two.

Here is an earlier question;

 

[haruspex]

For the annular slicing, isn't it just the annular integral for the hemisphere, but integrating from a non-zero smallest radius (namely...)?