Hello,

Problem:

I can't find my mistake when solving the surface area of a sphere.

Description of the sphere:

Given a sphere of radius $\displaystyle R$ centered at the origin. There is a circle at distance $\displaystyle y$ from the origin which has a radius of $\displaystyle h = \sqrt{R^2 - y^2}$ which is inscribed in the sphere

Description of the problem:

I was solving both the volume and area of a sphere. At first I calculated the volume of the sphere in the following approach:

$\displaystyle S_{circle} = \pi h^2 = \pi (R^2 - y^2 )$ (Area of the circle)

$\displaystyle V = \int_{-R}^{R} \pi (R^2 - y^2 ) dy = \frac{4}{3} \pi R^3$

Where it calculates the area of every circle and multiplicates by the differential. No problems here.

Now I wanted to do the same for the surface area, but using the circumference length instead, therefore:

$\displaystyle C_{circle} = 2 \pi h = 2 \pi \sqrt{R^2 - y^2}$ (Length of the circumference)

$\displaystyle S = \int_{-R}^{R} 2 \pi \sqrt{R^2 - y^2} dy = \pi ^2 R^2 \neq 4 \pi R^2$