How Do You Calculate the Area Between Two Overlapping Circles?

[balrog1212]

Hi,

I've been working on this problem for a few days and I'm pretty stuck. I'm working on how much area is eclipsed by some arbitrary planet by another over a given time. My question is if I have two circles one given by (y-100)^2+x^2 = 100^2 and the other given by (y+100)^2+x^2 = 100^2, how much area will be between those two curves as the second circle is shifted up until it is directly on top of the other circle?

my attempt to do this was to make the first circle (y-100)^2+x^2 = 100^2 and the second circle (y+100-t)^2+x^2 = 100^2. I would then want to find the area between those two curves as t goes from 0 to 200.

My ability to take integrals with multiple variables is poor but I will be taking calc 3 next semester so that should help.

any information on how to set this up or general help would be greatly appreciated.

Thanks

 

[mathman]

You should be able to do this with plane geometry. Overlap consists of two pieces of the circles, where each piece is a sector minus a triangle.

 

[balrog1212]

Could I just use plane geometry to find the area between the two circles through the whole process of t going from 0 to 200?

I guess I could rephrase my question as "If I wanted to find the area between the two circles as t goes from 0 to 200 what would it be?"

at t=0 the area would be zero, at t=200 the area would be 10,000pi

so the total area I want to find would be (area at t=0+...+area at t=50+...+area at t=100+...+area at t=150+...+area at t=200) with every area in between those accounted for too.

 

[mathman]

The basic calculation gets one half, so at the end it is doubled.

Let r = radius of circle, s = t/2 (makes life easier for me). Consider line (L) connecting the two points where the circles overlap. The line from circle center to L has length r-s.
L has length 2√(2rs-s²). Therefore area of triangle is:
A=(r-s)√(2rs-s²).

The arc of the circle overlap is B=2arccos(1 - s/r). Therefore the area of the sector is C=Br²/2.

So the area you want is 2(C - A).

I suggest you check my calculations, so you will understand what is going on.

 

[CellCoree]

Dear friend,

Thank you for reaching out for help with your problem. It seems like you have a good understanding of the setup for the problem and your approach is on the right track. To find the area between two curves, we can use the formula for the definite integral. In this case, we will need to integrate with respect to both x and y.

To set up the integral, we can start by finding the intersection points between the two circles. These points will determine the limits of integration for both x and y. Once we have those limits, we can use the formula for the definite integral to find the area between the curves.

Since you mentioned that your ability to take integrals with multiple variables is poor, I would recommend reviewing the concept of double integrals and practicing some problems before attempting this one. Also, as you mentioned, taking calc 3 next semester will definitely help with your understanding and ability to solve these types of problems.

If you need any further assistance, please don't hesitate to reach out. Good luck!

Best,