surface area of a sphere using cartesian coordinates
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?
Re: surface area of a sphere using cartesian coordinates
First, you're right, that square root thing is the arc length, not just the height. Second, you're sort of correct in that, as you let dx get real small, dy also gets real small, so that square root LOOKS LIKE it is resolving to just the square root of one, which gets you to your result. But while dy is getting small, dy RELATIVE TO dx may not be getting small at all. That's what dy/dx is. At the equator, dy is actually changing quite rapidly wrt dx, so dy/dx is rather large, and if you don't have the full square root thing in there, you'll miss a lot of the surface area.