I'm trying to derive the equation for the surface area of a sphere by using a one-dimensional integral in cartesian coordinates, but I keep getting the wrong answer. Here is my reasoning:
The circumference of a circle is 2*pi*r
For a circle in the x-y plane, the equation is x^2 + y^2 = r^2, or x = sqrt(r^2 - y^2).
If we let x, as defined above, be the radius of a circle (in the z-x plane), we can calculate the area of an element of the surface area as a ring, with circumference 2*pi*sqrt(r^2 - y^2)*dy . We should then be able to integrate with y going from -r to r and get the surface area of the sphere. However, evaluating this integral only yields pi^2 * r^2, not the right formula.
What is wrong with my method here. Am I making a wrong assumption?