Is the Triple Integral in Cylindrical Coordinates Correctly Solved?

[daphnelee-mh]

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

 

[anuttarasammyak]

\int_2^8 dz \int_0^{2z} 2\pi \rho^3 d\rho

 

[daphnelee-mh]

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

 

[anuttarasammyak]

Find attached my handwriting.

 

[Delta2]

\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]

Yes. Thanks for correction.

 

[daphnelee-mh]

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

 

[Delta2]

Yes it looks correct to me.

 

[daphnelee-mh]

Thank you so much

 

[haruspex]

View attachment 265726
Can you help to see whether this is correct ? Thank you

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

 

[daphnelee-mh]

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

?

 

[haruspex]

?

So 1008 is divisible by 3.

 

[daphnelee-mh]

So 1008 is divisible by 3.

okay , thank you