• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Understanding the argument of the surface area integral

212
15
1. The problem statement, all variables and given/known data

Find ##\iint_S ydS##, where ##s## is the part of the cone ##z = \sqrt{2(x^2 + y^2)}## that lies below the plane ##z = 1 + y##

2. Relevant equations


3. The attempt at a solution

I have already posted this question on MSE: https://math.stackexchange.com/questions/3155620/surface-integral-of-an-intersection-cone-plane/3155634?noredirect=1#comment6498746_3155634

My issue is with ##\iint_S ydS =\sqrt{3} \int_A ydxdy=\sqrt{3}\, \bar{y}|A|##. Concretely, I do not get why ##\bar{y}## shows up.

My issue is that I still do not understand how to deal with the argument of the surface integral. Let's say we had for instance ##\iint_S xydS## or ##\iint_S y^2dS##. I wouldn't know how to proceed. I know it is somehow related to the centroid of the figure (in this case an elliptical cylinder).

Robert Z provided a short explanation but I do not get it...

May you please provide an explanation?

Thanks

 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,462
714
I think I get what is confusing you. Let's say you have an ellipse for which you know the centroid (center). Say the center is at ##(c,d)## and the major and minor axes are ##a## and ##b## and you know the formula for the area of such an ellipse is ##\pi a b##. Now let's say you have an integral that you want to evaluate something like ##\iint_A y~ dA## over the elliptical area. That is going to be a bit of work, but you can save yourself the work because you know the formula for the ##y## centroid of area of a region is$$
\bar y = \frac {\iint_A y~dA}{\iint_A 1~dA}$$So instead of working your integral out just note you can solve this equation for ##\iint_A y~ dA## getting$$
\iint_A y~ dA = \bar y \cdot \text{Area of ellipse} = d \pi a b$$since ##d## is the ##y## coordinate of the center of the ellipse.
 
212
15
I think I get what is confusing you. Let's say you have an ellipse for which you know the centroid (center). Say the center is at ##(c,d)## and the major and minor axes are ##a## and ##b## and you know the formula for the area of such an ellipse is ##\pi a b##. Now let's say you have an integral that you want to evaluate something like ##\iint_A y~ dA## over the elliptical area. That is going to be a bit of work, but you can save yourself the work because you know the formula for the ##y## centroid of area of a region is$$
\bar y = \frac {\iint_A y~dA}{\iint_A 1~dA}$$So instead of working your integral out just note you can solve this equation for ##\iint_A y~ dA## getting$$
\iint_A y~ dA = \bar y \cdot \text{Area of ellipse} = d \pi a b$$since ##d## is the ##y## coordinate of the center of the ellipse.
Oh so if we were to have:

$$\iint_A yx~ dA$$

Could I do:

$$\iint_A yx~ dA = \bar y \bar x \cdot \text{Area of ellipse} = de \pi a b$$

?

I have just seen ##x##, ##y## and ##z## applied individually but not multiplying (that is why I am asking).
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,462
714
Oh so if we were to have:

$$\iint_A yx~ dA$$

Could I do:

$$\iint_A yx~ dA = \bar y \bar x \cdot \text{Area of ellipse} = de \pi a b$$

?

I have just seen ##x##, ##y## and ##z## applied individually but not multiplying (that is why I am asking).
You can probably answer that for yourself. You would be using a formula like this:$$
\bar x \bar y = \frac {\iint_A xy~dA}{\iint_A 1~dA}$$You know the formulas for ##\bar x## and ##\bar y##. Put them in there and see if you think it is true.
 
212
15
You can probably answer that for yourself. You would be using a formula like this:$$
\bar x \bar y = \frac {\iint_A xy~dA}{\iint_A 1~dA}$$You know the formulas for ##\bar x## and ##\bar y##. Put them in there and see if you think it is true.
OK so I think you may agree that:

$$\iint_A xy~ dA = \bar x \bar y \cdot \text{Area of ellipse} = cd \pi a b$$
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,462
714
OK so I think you may agree that:

$$\iint_A xy~ dA = \bar x \bar y \cdot \text{Area of ellipse} = cd \pi a b$$
You are just guessing. Until you show me what happens when you do what I suggested in the last line of post #4 you won't be able to do anything but guess. And you won't learn anything.
 
212
15
You are just guessing. Until you show me what happens when you do what I suggested in the last line of post #4 you won't be able to do anything but guess. And you won't learn anything.
We know that:

$$\bar x = \frac {\iint_A x~dA}{\iint_A 1~dA}$$


$$\bar y = \frac {\iint_A y~dA}{\iint_A 1~dA}$$

Then:

$$(\frac {\iint_A x~dA}{\iint_A 1~dA})(\frac {\iint_A y~dA}{\iint_A 1~dA}) = \bar x \bar y = \frac {\iint_A xy~dA}{\iint_A 1~dA}$$

So this equality doesn't hold. Then the following is incorrect:

$$\iint_A xy~ dA = \bar x \bar y \cdot \text{Area of ellipse} = cd \pi a b$$

Then we have no alternative but work the integral out.
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,462
714
We know that:

$$\bar x = \frac {\iint_A x~dA}{\iint_A 1~dA}$$


$$\bar y = \frac {\iint_A y~dA}{\iint_A 1~dA}$$

Then:

$$(\frac {\iint_A x~dA}{\iint_A 1~dA})(\frac {\iint_A y~dA}{\iint_A 1~dA}) = \bar x \bar y = \frac {\iint_A xy~dA}{\iint_A 1~dA}$$

So this equality doesn't hold. Then the following is incorrect:

$$\iint_A xy~ dA = \bar x \bar y \cdot \text{Area of ellipse} = cd \pi a b$$

Then we have no alternative but work the integral out.
Good, that's a step in the right direction. So it looks like the equality doesn't hold. You do understand that "looks like" isn't a mathematical argument though, right? So what you should do now to really settle the matter for yourself is actually prove that it doesn't hold by working out a simple example and showing you get different numbers. Then you will KNOW it doesn't hold.
 
Last edited:

Want to reply to this thread?

"Understanding the argument of the surface area integral" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top