Non translation polar Limits of integration for volume of cylinder not centered at origin

Question

where $D:=(x-1)^2+y^2≤1$,

How do we find $$ \iint_D dxdy?$$

I'm not sure how to find the volume of a cylinder not centered at the origin using a double integral with polar coordinates. I'm not sure how to setup the limits of integration.

Obviously, the volume of cylinder is independent of where we center it but are the bounds of integration the same as if the cylinder were centered at the origin?

Is the radius of this cylinder not centered at the origin two or one? What I tried:

Let $ (x-1) = cos(\theta), y= sin(\theta)$

Answer 1

Rewrite $(x-1)^2 + y^2 \leq 1$ as $x^2 + y^2 \leq 2x$, which gives $r^2 \leq 2r\cos\theta$, or $r \leq 2\cos\theta$ (for $\cos\theta$ positive)

Therefore, the region is $\theta$ from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$ (where cosine is positive) and $r$ from $0$ to $2\cos\theta$

Answer 2

The given inequality represents the interior of a circle with a radius 1 and centred at (1,0).

The given integral represents the area of this circle.

Limits of x will be from 0 to 2. Limits of y will be from -1 to 1.