Alright. I completely confused about determining the area between regions of polar curves. However, I do feel that I have a solid grasp in finding areas for single functions. For a given function in polar form, I know that I find the limits of integration by setting the function equal to zero and solving for those theta values. This is the area "swept out". When presented with two functions, I understand that I must set them equal to find points of intersection. I do not know how to apply these points of intersection. For two given curves, do I take the difference of the integrals from those two shared points?(adsbygoogle = window.adsbygoogle || []).push({});

Here is an example:

The region inside the rose r = 4sin2θ and r = 2.

Equation = 1/2 ∫α→β (f(θ)^2 - g(θ)^2)dθ

I set these two functions equal.

sin2θ = 1/2

2θ = pi/6 , 5pi/6...

θ = pi/12 , 5pi/12...

When I look at the graph of this function, I see that I want the areas inside the circle r = 2 that belong to the other function. I should calculate the area of one section and multiply it by 4 to find the total area. I do not see how I can compute the partial area for this by using those two points of intersection. There seems to be one point I am missing or something and I cannot seem to figure out how to find it. Am I missing a theta value for which sin = 1/2 ?Even if I had two points, I do not understand how the I would take difference of those two areas to find this. If I took the difference in area of r = 4sin2θ and r = 2 , I would have the area outside of the circle. If I did the opposite, I would have negative area.

Thanks for reading this and helping.

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# Limits of integration for regions between polar curves

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