Volumes concept (1 Viewer)

[porcupinetree]

Hi everyone,

I realise this is probably a simple concept which I just don't seem to be able to grasp very well.
This was the question which got me thinking about it: https://imgur.com/tqbIeIi

Basically, I'm confused about how I should understand calculating the volume in part (ii) in terms of what boundaries to place on the integral - especially as we added two 'areas' that were for different boundaries.
I did realise that in part (i), for the inner cylinder/shell, the radius of the shell would typically be 1+x when the boundaries are -1 to 0 (because the domain is entirely negative, hence 1+x is 1-|x|), and that this is equivalent to 1-x with boundaries 0 to 1, which gives the desired result. Hence we can use the boundaries 0 to 1 for the integral in part (ii)

But is there a general concept here that can be applied to other situations like this, too? How should I understand adding areas/volumes like this?

Thanks

 

[mrpotatoed]

If I am not mistaken, this is pappus's theom, where you do two shells at once, for each value of r. The left hand area has radius 1-x, and the right hand area had radius 1+x. The limits go from 0 to 1, as that takes care of both sides (as you said). You will most likely be stepped through the method if this type of question is asked, as in, the question will be split into multiple parts to help you set up the integral, so generally not to much to worry about... but the method you used to work through it is fine for these question types, from my knowledge.