Numerous times in middle school I thought finding the surface area of a sphere is a trivial matter with calculus: Simply, like the volume derivation, add up the lengths of the circumferences at each cross section.
If the sphere was centered at the origin, we can state the formula R(x) for finding the radius at some point x to be:
R(x) = SQRT(r^2 - x^2)
Thus circumference would be:
C(x) = 2pi*SQRT(r^2 - x^2)
Now we integrate like so:
\(\int_{-r}^{r}2{\pi}\sqrt{r^2-x^2} \ dx\)
\(2{\pi}\int_{-r}^{r}\sqrt{r^2-x^2} \ dx\)
This should be a trivial integral... We know that the integrand is actually just the equation of a circle. Integrating this within the interval [-r, r] finds the area of the circle. Thus, we have,
\(2{\pi}({\pi}r^2) = 2{\pi^2}r^2\)
Thus is the surface area of a circle, which is clearly wrong.
What happened?
