Iterated integral in cylindrical coordinates that may be used to find volume

[LoffieZA]

Hello, I am having some problems with defining the boundaries of the integral. I am not very confident in my answer. Please see the attached images for the question and my working.

Any help will be appreciated.

 

[LCKurtz]

Looks good. Nicely done. Only change I would have done is the [MATH]\theta[/MATH] integral as [MATH]\int_0^\pi[/MATH], but yours gives the same answer.