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Triple integral: finding volume(with attempt)

Attachment 26532

Plz have a look on the image I uploaded above to see my question.

I tried to solve the question, and come up with two thoughts.

I must have some wrong intuitions, I used to think method 1 and method 2 will have the same answer, but actually they don't.

Which of the integrals, dz, dr or dθ is incorrect?

and what is the correct boundary?

Re: Triple integral: finding volume(with attempt)

Hey happymatthematics.

There is a simpler method using rotation of volumes since z = b is aligned with the axes: do you have to use one technique over another?

Re: Triple integral: finding volume(with attempt)

hi chiro.

I've solved the problem with the technique of cylindrical coordinate(r,Θ,z).

but I don't know how to deal with this problem with spherical coordinate(ρ,Θ,φ).

could you desmonstrate how to do it in spherical coordinate? or tell me the range of φ and then let me try.

Thank you.