sodni3
2021-09-21
Answered

Use a double integral to find the area of the region. The region inside the circle ${(x-1)}^{2}+{y}^{2}=1$ and outside the circle ${x}^{2}+{y}^{2}=1$

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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.

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.

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.

Step 4

Our Region can now be defined.

Step 5

Set up the integrals

Step 6

Integrate with respect to r and run

Step 7

At this point the power reduction formula is necessary for further integration.

Step 8

Based on symmetry you can double the integral and run from 0 to pi/3 but I will show it without.

Step 9

Use trig to find the values of the inputs

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.

Result:

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