# computing area

• Jan 28th 2007, 10:32 AM
computing area
Hi,
I have this problem:
the surfaces r=2 and 4, $\theta=$30 degrees and 50 degrees, $\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.
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= $\sin\theta$ $r$ $dr$ $d\phi$

then integrate the according the limits and the total are should be
S= S1+S2+S3
Is my reasonnig correct?
Thank you
B
• Jan 28th 2007, 01:51 PM
galactus
It would appear it is a region enclosed by two concentric spheres.

${\rho}=2 \;\ and \;\ {\rho}=4$ are two spheres of respective radius 2 and 4.

$\Large\int_{\frac{\pi}{6}}^{\frac{5{\pi}}{18}}\int _{\frac{\pi}{9}}^{\frac{\pi}{3}}\int_{2}^{4}{\rho} ^{2}sin({\phi})d{\rho}d{\phi}d{\theta}$
• Jan 28th 2007, 09:26 PM
Galactus ,
thank you but I think that this is the question for the first question which is compute the volume enclosed.
The second question is the one that bothers me.
Quote:

2- Find the total area of the enclosed Volume.
B.
• Jan 29th 2007, 12:55 AM
ticbol
Quote:

Galactus ,
thank you but I think that this is the question for the first question which is compute the volume enclosed.
The second question is the one that bothers me.

B.

Your reasoning is correct, except that the differential of the vertical arc is r*sin(phi)*d(phi).
Not r*sin(theta)*d(theta).

So your d(S1), for the concave/convex surfaces at r=2 and r=4 each, should be
[r*sin(phi)*d(phi)]*[r*d(theta)]
= (r^2)(sin(phi))(d(phi))(d(theta))

Yours is (r^2)(sin(theta))(d(theta))(d(phi)).

[Well, unless, of course, your vertical angle is theta, and your horizontal angle is phi. If that is the case, then your d(S1) is correct. It's just not the conventional way of doing it.]

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Then, d(S2), for the "horizontal" walls, should be (r*d(theta))(dr) = r*dr*d(theta)

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And, d(S3), for the "vertical" walls, should be (r*sin(phi)*d(phi))(dr) = r*sin(phi)*d(phi)*dr.