I’m trying to remember my calculus and use it to derive the surface area of a sphere. To do that, I’m imagining it as a stack of rings of decreasing size, piled from height $y=0$ to height $y=r$. To calculate the radius of the ring at a given height $y$, I’m using the formula $\sqrt{r^2-y^2}$.
I’ve used this method to derive the volume of a sphere and it works fine. The area of each ring is $\pi \cdot r^2$, and substituting the above formula for the radius says that at a given $y$, the area $= \pi \cdot (r^2 – y^2)$. If you integrate this from $y=0$ to $y=r$, you get $\frac{2 \pi r^3}{3}$ which is half the volume of the sphere, which makes sense since we’re only counting the top half of the sphere.
However if you substitute the same radius formula for circumference, at a given $y$ the circumference $= 2 \pi \sqrt{r^2-y^2}$. Integrating that is somewhat trickier, but according to WolframAlpha it’s $\frac{\pi^2 r^2}{2}$, which if you double it for the 2 hemispheres is $\pi^2 r^2$ which is not correct for the surface area.
So I’m wondering, what’s wrong with this method?
Best Answer
This is the same as for the circumference of a circle. If you approximate it by a polygon, the perimeter will tend to the correct value. But if you approximate it by horizontal segments, the total length of the segments is always twice the diameter, and $\pi=2$ !