Area Integral for unknown portion of circle.

In summary, the problem requires an iterated integral for the area inside the circle r=2cos(theta) for -pi/2 < theta < pi/2 and outside the circle r=1 centered at the origin. To find the area in the overlap region, the two circles must be solved for their intersection points. The recommended approach is to integrate dA = r*dr*dtheta over the region -pi/3 < theta < pi/3 and 1 < r < 2*cos(theta), and the resulting area is close to 2pi/3.
  • #1
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My problem requires me to write an iterated integral for the area inside the circle r=2cos(theta) for -pi/2 <theta< pi/2 (circle of radius 1 centered at (1,0)) and outside the circle r=1 centered at the origin.

So I can write an iterated integral for the circle of radius 1 centered at (1,0) and then subtract the area portion of the circle at the origin. But what is this portion of area? Can I write another iterated integral for this area portion and then subtract it from the iterated integral? How would I approach this?

Any help would be greatly appreciated! Thanks!
 
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  • #2
So i was able to write an integral for the area of the circle, and tried integrating for the area between the curves of each circle. I am getting a value for the requested area (as described above) that seems unreasonable and do not know where I am making the mistake, or if I am evening doing it correctly. Any suggestions?
 
  • #3
The way I would do this would just be to find the area of the part of the circle centered at the origin which overlaps the other circle and then subtract that from the area of the circle centered at (1,0). To get the area in the overlap region you need to see where the two circles intersect, i.e. solve 2cos(theta) = 1. This should give you two values and then you can either integrate dA = r*dr*dtheta or you can just get the area geometrically since it is just a wedge of the circle. I think the answer should be 2pi/3.
 
  • #4
I think 2pi/3 would be the answer if the left side of the shape were bounded by straight lines as opposed to curved ones. Since this is an approximation to the area, I'll probably use it if I cannot figure something else out. Thanks so much for your help!
 
  • #5
I think this integral may work for the Area outside the circle, r=1 and inside the circle, r=2cos(theta)

polar coordinates:

The double integral (where 1 <= r <= 2cos(theta) and -pi/3 <= theta <= pi/3 ) of differential Area dA=rdrdtheta

This seems to make sense when visualizing the double integral.

Solving, I got Area = 1.91 which seems reasonable since it is about sixty percent of a circle of radius 1 and area pi.

Does this seem like a valid method?
 
  • #6
Ahh, you are right. Let me try again: I think you want to integrate dA = r*dr*dtheta over the region -pi/3 < theta < pi/3 and 1 < r < 2*cos(theta). This is becasue +-pi/3 are where the two circles intersect and for any fixed theta, the value of r in the region under consideration goes from the inner circle to the outer circle. When I evaluted this integral I got something which was fairly close to 2pi/3, as it should be.
 
  • #7
haha, you beat me to it by a minute! good work.
 

What is the formula for finding the area integral of an unknown portion of a circle?

The formula for finding the area integral of an unknown portion of a circle is A = πr²θ, where r is the radius of the circle and θ is the central angle in radians.

What is the purpose of calculating the area integral of an unknown portion of a circle?

The purpose of calculating the area integral of an unknown portion of a circle is to determine the area enclosed by the unknown portion of the circle. This can be useful in various mathematical and scientific applications.

How do you calculate the central angle in radians for the area integral of an unknown portion of a circle?

To calculate the central angle in radians, you need to know the fraction of the entire circle that the unknown portion occupies. This fraction is then multiplied by 2π to get the central angle in radians.

What are some real-life applications of the area integral for unknown portions of circles?

The area integral for unknown portions of circles has many real-life applications, including calculating the area of irregularly shaped objects, determining the area of land for agricultural purposes, and finding the area of overlap between two circles in engineering and construction.

Are there any limitations to using the area integral for unknown portions of circles?

One limitation of using the area integral for unknown portions of circles is that it assumes the circle is a perfect shape with a constant radius. In reality, many objects are not perfectly circular and may have varying radii, making this calculation less accurate. Additionally, this formula only works for circles and cannot be applied to other shapes.

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