Calculate double integral of the intersection of the ellipse and circle

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Discussion Overview

The discussion revolves around calculating the double integral of a function defined as f(x,y) = a0 + a1y + a2x + a3xy over the area of intersection between an ellipse and a circle. Participants explore various approaches to set up the integral, including the use of polar coordinates and the determination of intersection points.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how to calculate the double integral of the function f(x,y) within the intersected area of an ellipse and a circle, expressing uncertainty about the process.
  • Another participant notes that the equation of the circle depends on its center and radius, suggesting that the integration method will depend on how the two figures overlap.
  • A participant mentions that knowing the center of the circle and the ellipse, along with their dimensions, allows for the calculation of intersection points.
  • Suggestions are made to determine the x-values of the intersection points for the outer integral and to use the equations of the ellipse and circle to define the limits for the inner integral.
  • There is a proposal to separate the area for integration and a question about the use of polar coordinates, with a counterpoint suggesting that this might complicate the process.
  • One participant realizes that the y integral may be trivial if the integrand is 1, indicating a simpler approach to writing the x integrand as the difference between the equations for the circle and ellipse.

Areas of Agreement / Disagreement

Participants express differing views on the use of polar coordinates and the necessity of separating the area for integration. There is no consensus on the best approach to take for calculating the double integral.

Contextual Notes

Participants have not fully resolved the mathematical steps involved in determining the intersection points or the implications of using polar coordinates versus Cartesian coordinates.

zs96742
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how to calculate the double integral of f(x,y) within the intersected area?

f(x,y)=a0+a1y+a2x+a3xy

The area is the intersection of an ellipse and a circle.

attachment.php?attachmentid=54378&stc=1&d=1357046899.jpg


Any help will be appreciated, I don't know how to do this.

can I use x=racosθ,y=rbsinθ to transformer the ellipse and circle?
If I can, what's the equation for the circle?
what's the range for r while calculating.
 

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Well, it's your circle! I don't see how anyone else can tell you what its equation is. The equation of a circle with center at (x_0, y_0) and radius r is, of course, (x- x_0)^2+ (y- y_0)^2= r^2. How to integrate depends on exactly how the two figures overlap. x= r cos(\theta), y= r sin(\theta) assumes that r is measured from (0, 0). That might be useful if you were to "translate" your system so that the center of the circle is at the origin.
 
HallsofIvy said:
Well, it's your circle! I don't see how anyone else can tell you what its equation is. The equation of a circle with center at (x_0, y_0) and radius r is, of course, (x- x_0)^2+ (y- y_0)^2= r^2. How to integrate depends on exactly how the two figures overlap. x= r cos(\theta), y= r sin(\theta) assumes that r is measured from (0, 0). That might be useful if you were to "translate" your system so that the center of the circle is at the origin.

Yes, I forgot this, the center of the circle and the ellipse, the radius, the a and b are all known. That means I can calculate the intersection points.

If you chose the circle center as the origin, what's the equation for the ellipse?
 
From the picture and your previous remark, I suggest the following.

Get the x values of the intersection points - outer integral in x.

Use the y = for the ellipse as the lower and the y = for the circle as the upper limit - inner integral in y.
Since y is defined as a square root for both figures, make sure you pick the correct sign.
 
Last edited:
mathman said:
From the picture and your previous remark, I suggest the following.

Get the x values of the intersection points - outer integral in x.

Use the y = for the ellipse as the lower and the y = for the circle as the upper limit - inner integral in y.
Since y is defined as a square root for both figures, make sure you pick the correct sign.

In this case, I have to separate the area, any chance I can use polar coordinate?
 
zs96742 said:
In this case, I have to separate the area, any chance I can use polar coordinate?

What is the need to separate? It seems to me polar coordinates make it more complicated.

I also realized later, the y integral is trivial, since the integrand is 1. All you need to do is write the x integrand as the difference between the equation for y of the circle piece minus the equation for the ellipse piece.
 

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