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Integrating on polar plane with polar bears

تاریخ:پنجشنبه 1 بهمن 1394-08:14 ب.ظ

Hi
After a long time of inactivity, Now I'm back and decided to bring useful contents to you. Thank you
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Today, I want to give you some information about finding areas and certainly length of curves(I will talk about it in later posts) in polar coordinates
As we know, there is a close relation between Cartesian and polar coordinates. We integrate in Cartesian coordinates by choosing arbitrary intervals over the x axis and calculate the sum of areas of rectangles created between the function itself and the x axis. Note that elements of the area were infinite thin rectangles. That is supposed to be known here
The difference here is that the area which we want to calculate, is the area trapped between the function r=f(θ) and two origin-passing lines θ=a and θ=b.See figure 1Figure 1
Figure 1
A reliable way to integrate is to take origin-centered sectors with central angle of dθ as area elements. So by knowing the area of a sector by its radii and central angle and integrating it over the interval θ=a to θ=b, our job is done. Let's see
dS=1/2 r2 dθ
Note that r is a function of θ.Hence
dS=1/2 f2(θ) dθ
We can calculate the area by integrating both sides of the equation above over the interval θ=a to θ=b
Mission Done







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