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9.9 Finding the Area of the Region Bounded by Two Polar Curves

1 min readβ€’june 8, 2020

Sumi Vora

Sumi Vora


AP Calculus AB/BC ♾️

279Β resources
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Once you get the hang of finding the area under one curve, finding the area between two curves is pretty simple. Remember from previous units that when you find the area between two curves, you subtract the bottom curve from the top curve. This is the same in polar functions, but instead of subtracting β€œtop minus bottom,” you’ll subtract β€œouter minus inner.” 
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If the curves intersect, then you may have to find the area inside the curves by splitting the region.Β 
Example: Let R be the region inside the graph of the polar curve r = 4 and r = 4 + 2sin(2ΞΈ) on [0, Ο€]. Find the area of R.
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Since the graphs intersect at ΞΈ = 2 we can see that when ΞΈ < Ο€/2, r = 4 is on top, and when ΞΈ > Ο€/2, r = 4+2sin(2ΞΈ) is on top. Based on this information, we can construct two integrals:
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Polar Arc LengthΒ 

There is one last thing you need to know about polar functions: arc length. Finding arc length is pretty straightforward, but you do need to have the formula memorized for the exam.Β 
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