Surface area via integration confusion

Nov 2015
198
25
United States of America
Hello forum,

I am struggling on this problem.

I am asked to evaluate
$\oint \vec r \dot \,d\vec \sigma$​
over the whole surface of the cylinder bounded by
$x^2+y^2=1, z=0, z=3$​

It seems pretty straight forward geometrically as it is just a unit circle at $z=0$ and then it extends up to $z=3$ forming a cylinder.

Initially, I parametrized the unit circle as $\langle cos(t), sin(t)\rangle$ and

$d\vec \sigma$ pointing normal in the positive z direction.

This did not go anywhere however. Now I am thinking I should just work in cylindrical coordinates, but I am just getting kind of confused working with this. Does anyone have any tips for me?

Thank you always!
 
Nov 2015
198
25
United States of America
I should add...

I did $\iint_A \,dA \Rightarrow \int_{0}^{3} dz \int_{0}^{2\pi} d\theta + 2\pi$ = $6\pi + 2\pi = 8\pi$

However, I did this using different techniques from what was asked in the problem. The $2\pi$ added at the end is to account for the "top circle" of the cylinder.
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
Because the figure is not "smooth", this has to be done in three parts:
1) The base at z= 0. This is a disk with center at (x, y)= (0, 0). We can take "\(\displaystyle d\vec{\sigma}\)" to be \(\displaystyle <0, 0, -1> dxdy\) or, in polar coordinates, \(\displaystyle <0, 0, -1>r drd\theta\) ("-1" because it is pointing downward. Integrating \(\displaystyle \vec{r}\), which, I assume, is \(\displaystyle <x, y, 0>\)(?), That gives \(\displaystyle \int_0^{2\pi}\int_0^1 r^2 drd\theta\).

2) The base at z= 3. Now \(\displaystyle d\vec{\sigma}= \,<0, 0, 1>dxdy\) or \(\displaystyle d\vec{\sigma}= \,<0, 0, 1>r dr d\theta\). The integral is the same as above. The difference between the "-1" and "1" is irrelevant because the integrand is independent of z.

3) The curved sides. Yes, you can take \(\displaystyle x= cos(\theta)\), \(\displaystyle y= \sin(\theta)\), z= z. \(\displaystyle \vec{d\sigma}= \,<\cos(\theta), -\sin(\theta), 0>d\theta dz>\). The integrand, \(\displaystyle \vec{r}= \,<\cos(\theta), -\sin(\theta), 0>\) so the integral becomes \(\displaystyle \int_0^3\int_0^{2\pi} \cos^2(\theta)+ \sin^2(\theta) d\theta dz= \int_0^3\int_0^{2\pi} d\theta dz\)
 
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Nov 2015
198
25
United States of America
Because the figure is not "smooth", this has to be done in three parts:
1) The base at z= 0. This is a disk with center at (x, y)= (0, 0). We can take "\(\displaystyle d\vec{\sigma}\)" to be \(\displaystyle <0, 0, -1> dxdy\) or, in polar coordinates, \(\displaystyle <0, 0, -1>r drd\theta\) ("-1" because it is pointing downward. Integrating \(\displaystyle \vec{r}\), which, I assume, is \(\displaystyle <x, y, 0>\)(?), That gives \(\displaystyle \int_0^{2\pi}\int_0^1 r^2 drd\theta\).

2) The base at z= 3. Now \(\displaystyle d\vec{\sigma}= \,<0, 0, 1>dxdy\) or \(\displaystyle d\vec{\sigma}= \,<0, 0, 1>r dr d\theta\). The integral is the same as above. The difference between the "-1" and "1" is irrelevant because the integrand is independent of z.

3) The curved sides. Yes, you can take \(\displaystyle x= cos(\theta)\), \(\displaystyle y= \sin(\theta)\), z= z. \(\displaystyle \vec{d\sigma}= \,<\cos(\theta), -\sin(\theta), 0>d\theta dz>\). The integrand, \(\displaystyle \vec{r}= \,<\cos(\theta), -\sin(\theta), 0>\) so the integral becomes \(\displaystyle \int_0^3\int_0^{2\pi} \cos^2(\theta)+ \sin^2(\theta) d\theta dz= \int_0^3\int_0^{2\pi} d\theta dz\)
Thanks for clearing up my confusion!
 
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topsquark

Math Team
May 2013
2,510
1,049
The Astral plane
I'm going to add an extra comment here. It has no direct bearing on how to solve the current problem, which has been nicely done. However...

When putting this problem into cylindrical coordinates, you really should make the substitution \(\displaystyle x = r ~ \cos( \theta )\) and \(\displaystyle y = r ~ \sin( \theta )\) which has a differential area element \(\displaystyle r~dr~d \theta \).

In this case, we are looking at the cylinder bounded by \(\displaystyle x^2 + y^2 = 1\) so r = 1 anyway. But I think it is a valuable comment, as this will not always be the case.

-Dan
 
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