Surface of enclosed volume

[brad sue]

Homework Statement

Hi,
I have this problem:
the surfaces r=2 and 4, \(\displaystyle \theta=\)30 degrees and 50 degrees, \(\displaystyle \phi=\)20 degrees and 60 degrees identify a closed surface.
1- find the enclosed volume.
2- Find the total area of the enclosed surface. ( I think it is a typo from the teacher. It is volume not surface)

The first question is straigth forward
For the secon question I have some issues.

Homework Equations

Do I need to take take each element of surface (in spherical coordinates)
dS1=$$r^2$$ sin(theta) $$d\theta$$ $$d\phi$$

dS2=$$r$$*$$dr$$$$d\phi$$
dS3=$$\sin\theta$$*$$r$$$$dr$$$$d\phi$$

The Attempt at a Solution

integrate the according the limits and the total are should be
S= S1+S2+S3
Is my reasonnig correct?
Thank you
B

 

[mighty2000]

ella

Hello Bella,

Yes, your reasoning is correct for finding the enclosed volume. To find the total surface area, you will need to integrate over each element of the surface, which can be calculated using the equations you provided. However, you will also need to take into account the different limits for each surface element. For example, for the surface element dS1, the limits for \theta and \phi will be 30 degrees to 50 degrees and 20 degrees to 60 degrees, respectively. You will need to set up multiple integrals for each surface element and then add them together to get the total surface area. I hope this helps! Good luck with your calculations.