Change of variables - Double integrals

Question

I have trouble understanding how the limits work regarding polar coordinates in a double integral. For example, say if I had the equation $$(x-2)^2 + y^2 = 1.$$ This is a circle centred at (2,0) with radius 1, so the obvious change of variable is to polar coordinates. Now, $\theta$ varies from $0$ to $2\pi$, but is $r$ the distance from the origin or the radius of the circle in this case? Are the limits from $1$ to $3$ or $0$ to $1$?

Answer 1

The technical way to convert to polar is to use the substitution $x=r\cos\theta$, $y=r\sin\theta$. Then you would get

$$(r\cos\theta-2)^2+r^2\sin^2\theta=1$$

It may be easier, however, to use another substitution first: $t=x-2$.

Then we have $t^2+y^2=1$, or $r=1$ in polar coordinates.