Let #b > a > 0# be constants. Find the area of the surface generated by revolving the circle #(x − b)^2 + y^2 = a^2# about the y-axis?

To find the area of the surface generated by revolving the circle ((x - b)^2 + y^2 = a^2) about the y-axis, we can use the formula for the surface area of revolution:

[ A = 2\pi \int_{c}^{d} y \sqrt{1 + \left(\frac{dx}{dy}\right)^2} , dy ]

First, solve the equation of the circle for (x) to get (x = \pm \sqrt{a^2 - y^2} + b).

Then, differentiate (x) with respect to (y) to find (\frac{dx}{dy}).

Next, integrate (y \sqrt{1 + \left(\frac{dx}{dy}\right)^2}) with respect to (y) from (c) to (d), where (c) and (d) are the (y)-coordinates of the points of intersection of the circle and the y-axis.

Finally, multiply the result by (2\pi) to get the total surface area.

Please note that the limits of integration and other details would need to be determined based on the specific circle and the points of intersection with the y-axis.