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I was trying to derive the formula for the surface area of the curved part of a cone, but ran into a problem. Let $f(x)=kx$ be any line with any angle passing through the origin, $h$ be the height (non-slanted) of the cone, $l$ be the slant height, and $r$ be the base radius.

Then, the surface area of the cone should be the sum of all of the circumferences of the values of $f(x)$ for $0\leq x\leq h$

A graph to show the line that will be surface area of revolutionized, h=1.5 and k=0.5

$$2\pi \int_{0}^{h} f(x)dx = 2\pi \int_{0}^{h} kx dx = \pi kh^2$$

From the graph, k is just the rise over the run: $\frac{r}{h}$

$$\pi kh^2=\pi rh$$ However, looking online the correct formula is $\pi rl$, and I am confused where I went wrong.

Evgeny
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Consider slicing the cone into many thin coin-like slices. You want to set up the integral that corresponds to adding up the area of the "side wall" of each slice. Which is to say, each slice has a thin circular strip of area that you're after.

You're not taking into account the fact that the sides of the slices are slanted. So for each slice, the width of the strip that you are trying to calculate the area of isn't $\Delta h$, it is $\Delta l$.

Arthur
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  • Oh wow, that makes a lot of sence. I thought of integration as the sum of infintesimal steps, so I did not think that the slant of it would affect this. A follow up question however, if I did rotate the line y=kx and got a shape, its not a cone right? But what is it? – Evgeny Mar 02 '22 at 11:46
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    @Evgeny If you rotate the line $y = kx$ (the line in your Desmos drawing) about the $x$-axis, then yes, you get a cone. And yes, it can be a bit difficult to obtain intuition on what matters and what doesn't in an integration like this. It does not matter, for instance, that each slice is a little narrower on one end than the other (in that the circumference changes from one end of the slice to the other). This would technically make the area calculation for each slice a bit more difficult, but it may safely be ignored. – Arthur Mar 02 '22 at 11:51
  • Thanks for the explanation, because I was wondering, if you have a slice where you consider $\Delta l$ it will end up in a "slanted" rectangle, so I guess that "slant" can be safely ignored unlike the "slant" of the cone ($\Delta l$ instead of $\Delta h$) – Evgeny Mar 02 '22 at 14:26