Find the area of the region that lies between both curves.&nbsp;r2=50sin⁡(2θ),&nbsp;r=5

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Find the area of the region that lies between both curves.&amp;nbsp;r2=50sin⁡(2θ),&amp;nbsp;r=5

Otigh1979 · Accepted Answer

Step 1  Consider the following curves:  r2=50sin⁡(2θ), r=5  The objective is to find the area of the region that lies inside both the curves.  Sketch the region as follows:    Find the intersection points as follows:  From the curves,  50sin⁡2θ=52  sin⁡2θ=2550  sin⁡2θ=12  2θ=π6, 5π6, 13π6, 17π6  θ=π12, 5π12, 13π12, 17π12  The formula for the area A of the polar region R is,  A=∫αb12[f(θ]2dθ  Step 2  The desired area can be found by adding the area inside the cardioid between θ=0 and π4 from the area inside the circle from 0 to π4  Since the region is symmetric about θ=π4, you can write the area is,  A=2{2×12∫0π12(50sin⁡(2θ))2dθ+2×12∫π12π452dθ}  =2{∫0π1250sin⁡(2θ)dθ+∫π12π425dθ}  =2{[−502cos⁡2θ]0π12+25[θ]π12π4}  =2[−25(cos⁡π6−cos⁡0)+25(π4−π12)]  =2[−25(32−1)+25(2π12)]  =2(−2532+25

Find the area of the region that lies inside both

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