How Do I Apply the Theorem of Pappus to Find the Volume of a Revolved Region?

[blumfeld0]

The theorem of pappus seems too simple that i do not get it.
ok
suppose I have the 1. (x cm., y c.m) meaning i have the x and y center of mass (the centroids). the center of mass is of some function
2. y(x) how do i find the volume when i revolve this region about the line 3. y(x) = mx+b

thank you

 

[HallsofIvy]

No, the center of mass is not "some function". The center of mass is a single point: of some 2 dimensional region you have not mentioned.

And since you haven't mentioned the region, you haven't mentioned the area of the region which is the the cross-sectional area of the volume swept out. Pappus' theorem says that if you have a region of area A, rotated about a line, the volume of the figure created is 2$\pi$RA where R is the distance from the centroid of the region.

Calculate the distance, R, from the centroid of the given region to the line y= mx+ b, calculate the area of the region, and then use $V= 2\pi RA$.

 

[blumfeld0]

Ok here is a sample of what I mean.
Given the centroid of the region enclosed by the x-axis and
y = Sqrt[a^2-x^2] (a semicircle) is located at (0,4a/3Pi).
find the volume of the solid generated by revolving this region about the y= x-a.

So I know Volume = 2Pi*area*distance from centroid to line y=x-a

Area = Pi * a^2 right?
distance to from centroid to line y = x-a is what??

thank you

 

[HallsofIvy]

The distance from a point, $(x_0, y_0)$, to a line Ax+ By+C= 0 is given by
[tex]\frac{|Ax_0+ By_0+ C|}{\sqrt{A^2+ B^2}}[/itex]<br /> <br /> The area of a <b>circle</b> is $\pi r^2$. The area of a <b>semi</b>- circle is half that.[/tex]

 

[blumfeld0]

Hi Thank you very much. quick question what does
|Ax +By+c| mean?
I do not understand what the tabs means?

can you please give me a quick example.
In my case

y= x-a
so |1x-1y-a|
does this equal x^2-y^2-a^2"
thank you

 

[HallsofIvy]

Hi Thank you very much. quick question what does
|Ax +By+c| mean?
I do not understand what the tabs means?

can you please give me a quick example.
In my case

y= x-a
so |1x-1y-a|
does this equal x^2-y^2-a^2"
thank you

| | is absolute value. But let's back up a little. WHY are you doing this problem? You don't know Pappus' theorem, you got the wrong area for a semi- circle and you have the wrong value for the centroid! If you are taking a multi-variable Calculus course, you should know all those things before you attempt a problem like this. I recommend you go back, re-read the problem, take a deep breath, and start all over again.

 

[blumfeld0]

We skipped this stuff when I was in calc II and I really want to get it so I am doing a couple of problems for fun. I know the area of a semicircle.
The problem says
"the centroid of the region enclosed by the x-axis and the semicircle
y= Sqrt[a^2-x^2] lies at the point (0,4a/3Pi). Find the volume of the solid generated by revolving this region about the line y = x-a"

Volume = 2Pi A *b(distance)

A= Pi*a^2/2
b = (1x0-1y0-a)/Sqrt[(1+1)] = |0-(4a/3Pi)-a|/Sqrt[2] =

(2*Pi*Pi*a^2/2) * ((4a/3Pi)+a) /Sqrt[2]

is that right?
thank you!