TITLE: Setting Up an Integral to Find A Cone's Surface Area
QUESTION [24 upvotes]: I tried proving the formula presented here by integrating the circumferences of cross-sections of a right circular cone:
$$\int_{0}^{h}2\pi sdt, \qquad\qquad s = \frac{r}{h}t$$
so
$$\int_{0}^{h}2\pi \frac{r}{h}tdt.$$
Integrating it got me $\pi h r$, which can't be right because $h$ isn't the slant height. So adding up the areas of differential-width circular strips doesn't add up to the lateral surface area of a cone?
EDIT: I now realize that the integral works if I set the upper limit to the slant height - this works if I think of "unwrapping" the cone and forming a portion of a circle. The question still remains though: why won't the original integral work? Won't the value of the sum of the cylinders' areas reach the area of the cone as the number of partitions approaches infinity?

REPLY [6 votes]: The fundamental issue with using the integral stated in the original question is that is the wrong shape to use.  It is using small cylinders to approximate what are in reality frustums (except the first one, which is a small cone).  As the cylinders become smaller and smaller, as we approach the limit of the height of each becoming 0, those heights remain proportionally smaller than the slant height of the frustums.  As those heights get smaller, the number of cylinders proportionally increases.  When you sum it all up, the error between the approximated surface area to the actual surface area, remains unchanged.  
The easiest way to see this is to approximate through numerical integration.  Go ahead and start with one cylinder to approximate the cone.  If you use the average radius for the cone, you will get a total surface area = $\pi h r$.  Then split it into $2$ cylinders and add it up. You will get the same value for the total surface of the two cylinders.  As you continue to break it into smaller and smaller cylinders, the total surface area remains constant.  It remains the value of the integral stated above.  In essence, this is similar to multiplying too numbers together, where one number of approaching 0 and the other is approaching infinity and the rates perfectly offset each other.