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.
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.
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.
Our Region can now be defined.
Set up the integrals
Integrate with respect to r and run
At this point the power reduction formula is necessary for further integration.
Based on symmetry you can double the integral and run from 0 to pi/3 but I will show it without.
Use trig to find the values of the inputs
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.
Five circles are placed in a rectangle as shown. If the length of the shorter side of the rectangle is 1, find the length of the other side.