Use a double integral to find the area of the

sodni3 2021-09-21 Answered
Use a double integral to find the area of the region. The region inside the circle (x1)2+y2=1 and outside the circle x2+y2=1
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Expert Answer

tabuordg
Answered 2021-09-22 Author has 99 answers

Step1
A basic understanding of equations tells us that this equation is a circle shifted one unit to the right. I noticed that visualizing this in polar can be difficult so I laid out ever step for those that like to hand-jam.
(x1)2+y2=1
(rcosθ1)2+r2sin2θ=1
(r2cos2θ2rcosθ+1)+1r2sin2θ=1
r2cos2θ+r2sin2θ=2rcosθ
x2+y2=2rcosθ
r2=2rcosθ
r=2cosθ
Step2
Again, an understanding that this is the equation of a circle with radius 1 is useful, but if that cannot be seen you can convert to polar.
x2+y2=1
r2=1
r=1
Step3
Graphing this is simple and recommended to better understand the limits of integration. To find your limits of integration for r make the appropriate substitution and solve for theta.
1=2cosθ
cosθ=12
θ=π3,π3
Step 4
Our Region can now be defined.
R=(r,θ)1r2cosθ,π3θπ3
Step 5
Set up the integrals
π3π312cosθrdrdθ
Step 6
Integrate with respect to r and run
π3π312r212cosθdθ
Step 7
At this point the power reduction formula is necessary for further integration.
π3π32cos2θ12dθ
Step 8
Based on symmetry you can double the integral and run from 0 to pi/3 but I will show it without.
π3π32cos2θ+12dθ
Step 9
Use trig to find the values of the inputs
12sin2θ+12θ{π3}π3
Step10
I went more in detail with the set up of the problem than I did with the integration because I believe that's where the most confusion comes from. Please let me know if there are any problems with this solution.
π3+32
Result: π3+32

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