Expert: Paul Klarreich - 2/22/2009

Question
QUESTION: Consider the thin plate, of constant density p , which lies above the x-axis and between the two ellipses x^2+y^2/4=1 and x^2/4+y^2/16=1 Find its center of mass(HINT: In order to reduce the amount of integration required, use symmetry and the formula which states that the area A of the region enclosed by the ellipse A = piab )

ANSWER: Questioner:   nyasha machoko
Category:  Calculus
Private:  No
 
Subject:  Area of a an ellipse
Question:  Consider the thin plate, of constant density p , which lies above the x-axis and between the two ellipses x^2+y^2/4=1 and x^2/4+y^2/16=1 Find its center of mass(HINT: In order to reduce the amount of integration required, use symmetry and the formula which states that the area A of the region enclosed by the ellipse A = piab )
...............................

x^2 + y^2/4 = 1

4x^2 + y^2 = 4

a = 1,  b = 2
................
x^2/4 + y^2/16 = 1

4x^2 + y^2 = 16

a = 2,  b = 4
.......................
y[lower] = sqrt(4 - 4x^2)

y[upper] = sqrt(16 - 4x^2)



It seems to me that you will reduce the amount of integration almose to zero.  The inner half-ellipse, with area  pi(1)(2)/2 = pi is contained in the outer one, with area  pi(4)(2)/2 = 4pi.

So the total area (mass) = 3pi.

Now to the centroid, you want the y-coordinate only.  (The -x-coordinate is zero, by symmetry, as the problem reminds you.)

You want

{
| y dA
}

Now dA = (y[upper] - y[lower]) dx, for each vertical 'slice'.
And  y = median y for the slice, which is (1/2)(y[upper] + y[lower])

FOR THE part up to the end of the inner one,  at  x = 1.

Now here is what you must do:

To get the green area:

{
| y dA =
}

{1
| (1/2) ( y[upper] + y[lower] )( y[upper] - y[lower] ) dx =
}0

Now the sum-and-difference algebra should get rid of the roots and make for an easy integral.


To get the yellow area:

{
| y dA =
}

{2
| (1/2) ( y[upper] )( y[upper] ) dx =
}1

Again, it should be easy.

Good luck.



---------- FOLLOW-UP ----------

QUESTION:  pi(1)(2)/2 = pi

How come when calculation for the area you divided by two. Isn't the area of an ellipse =Pi * A * B

ANSWER: because you want the upper part only.

---------- FOLLOW-UP ----------

QUESTION: So when we do the integration for the green area aren't we supposed to multiply by two since we are integrating from 0 to 1 instead of integrating from -1 to 1.  

Answer
OK.  You are probably correct.    Keep in mind that your computation of the centroid looks like this:

         Integral with an extra factor of y.
y[centroid] = ----------------------------------------------------
         Integral without that factor of y, which is the area.

So if you do the top from 0 to 1 instead of -1 to 1, then you will do the bottom the same way, so take half the area. (or multiply by 2)