Solving this triple integral

Homework Statement:

(question is attached below)

Relevant Equations:

x^2+y^2=r^2
dV=rdrdzdΘ

I am trying to solve it using cylindrical coordinates, but I am not sure whether the my description of region E is correct, whether is the value of r is 2 to 4, or have to evaluate the volume 2 times ( r from 0 to 4 minus r from 0 to 2), and whether is okay to take z from r^2/2 to 8

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anuttarasammyak
Gold Member
$$\int_2^8 dz \int_0^{2z} 2\pi \rho^3 d\rho$$

Delta2
May I know which method did you use? I have no idea of it, thanks

anuttarasammyak
Gold Member
Find attached my handwriting.

Attachments

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daphnelee-mh
Delta2
Homework Helper
Gold Member
$$\int_2^8 dz \int_0^{2z} 2\pi \rho^3 d\rho$$
Shouldn't the boundary of integration be ##\sqrt{2z}## instead of ##2z##?

anuttarasammyak
anuttarasammyak
Gold Member
Yes. Thanks for correction.

Delta2

Can you help to see whether this is correct ? Thank you

Delta2
Homework Helper
Gold Member
Yes it looks correct to me.

Thank you so much

haruspex
Homework Helper
Gold Member
View attachment 265726
Can you help to see whether this is correct ? Thank you
The digits of 1008 add to 9, so...

The digits of 1008 add to 9, so...
?

haruspex