Setting up an integral for surface area

[Abyssnight]

Homework Statement

Set up an integral that represents the area of the surface generated when the region is bounded by x + y² = 1 and the y-axis, then rotated about the y-axis. All in one-variable.

Homework Equations

SA = 2pi \int xds
possible x² + y² = r²

The Attempt at a Solution

I tried to set up an integral as a circle and somehow ended up getting 2pi\int(r/2)(2pi r) dr

When I picture it, it would be a sphere with a radius of 1. Still don't truly understand on how to make a integral to show it

 

[Abyssnight]

I looked at it again
would one possible answer be: 4pi\int(-y^2 + 1)\sqrt{1 + 4y^2}dy
with limits of integration (0,1)