Calculate double integral of the intersection of the ellipse and circle

In summary, the conversation discusses the calculation of a double integral for f(x,y) within the intersected area of an ellipse and a circle. The equation for the circle is (x-x_0)^2 + (y-y_0)^2 = r^2 and the equation for the ellipse can be found by solving for the intersection points. The conversation also mentions the use of polar coordinates but it is deemed more complicated. It also notes that the y integral is trivial and the x integrand can be calculated by finding the difference between the equation for y of the circle piece and the ellipse piece.
  • #1
zs96742
5
0
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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  • #2
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 [itex](x_0, y_0)[/itex] and radius r is, of course, [itex](x- x_0)^2+ (y- y_0)^2= r^2[/itex]. How to integrate depends on exactly how the two figures overlap. [itex]x= r cos(\theta)[/itex], [itex]y= r sin(\theta)[/itex] 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.
 
  • #3
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 [itex](x_0, y_0)[/itex] and radius r is, of course, [itex](x- x_0)^2+ (y- y_0)^2= r^2[/itex]. How to integrate depends on exactly how the two figures overlap. [itex]x= r cos(\theta)[/itex], [itex]y= r sin(\theta)[/itex] 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?
 
  • #4
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.
 
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  • #5
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?
 
  • #6
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.
 

Related to Calculate double integral of the intersection of the ellipse and circle

1. What is a double integral?

A double integral is a mathematical tool used to calculate the area under a two-dimensional surface. It is represented by two integrals, one nested inside the other, and is used to find the volume of a three-dimensional shape.

2. How do you calculate a double integral?

To calculate a double integral, you first need to set up the limits of integration for both variables. Then, you integrate the inner integral with respect to one variable, while treating the other variable as a constant. Finally, you integrate the resulting function from the first integral with respect to the other variable.

3. What is the intersection of an ellipse and a circle?

The intersection of an ellipse and a circle is the set of points that lie on both the ellipse and the circle. It forms a closed curve that resembles a figure eight.

4. How do you find the intersection of an ellipse and a circle?

To find the intersection of an ellipse and a circle, you can set up a system of equations using the equations of the ellipse and the circle. Then, you can solve for the values of the variables that satisfy both equations, which will give you the coordinates of the points of intersection.

5. Why is it important to calculate the double integral of the intersection of an ellipse and a circle?

Calculating the double integral of the intersection of an ellipse and a circle can be useful in various applications in physics, engineering, and other fields. It can help determine the volume of a three-dimensional shape or the area of a two-dimensional region, which can be used in various calculations and analyses.

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