Vector Calculus - Cylindrical Co-ordinates (HELP PLEASE)

I understand the basis, I just dont understand how to get the individual limits for each integral for each question. Example 4.11a) I got z=sqrt((1-r^2)/3) to -((1-r^2)/3) and R=1 to -1 and theta=2pi to 0 but upon solving that, in the second integral r=1 and r=-1 cause the integral to become 0. Hence why I realise I have made a mistake.

Yeah that's the thing with solving integrals. The limits are correct but doing 2 * the integral with the limits r=0 and r=1 would get an answer that's not 0

So if I did the integral with all bottom limits=0 and taking the scale factor 2 out for both top and bottom half of the ellipsoid I would get the answer as 4pi/3root3. Does that sound okay to you?

No, just the one integral involving -1/1 requires the 0/1, scale by two treatment

I got 8pi/3root3, does that sound right to you?

I( could help you but I am teaching all day

Try this link

http://madasmaths.com/archive_maths_booklets_advanced_topics.html

download file

multivariable_integration_in_polars
( long download time)
there are several cylindrical polar questions with solutions towards the end

See if it helps

Afraid not sorry, I did read them through.

So if I did the integral with all bottom limits=0 and taking the scale factor 2 out for both top and bottom half of the ellipsoid I would get the answer as 4pi/3root3. Does that sound okay to you?