Consider the function $y=y(x)$. We can rotate the graph of this function around the x-axis to get a surface of revolution. The area of this surface of revolution between $x_0$ and $x_1$ is given by:
$$\int_{x_0}^{x_1} 2 \pi y \sqrt{1+y'}dx$$
The arc length of the graph of our function between the same bounds is
$$\int_{x_0}^{x_1}\sqrt{1+y'}dx$$
Naively, one might incorrectly think that $$\int_{x_0}^{x_1} 2 \pi ydx$$ will give us the surface area. Can somebody explain to me, with words, in the least technical and most intuitive way possible, why exactly this last integral is incorrect? What exactly is it calculating? Why is the arc length term necessary?
My thoughts: A part of me feels like the arc length term should be unneccessary, because as the limit of width of the rectangles in the Riemann sum goes to $0$, any information about the steepness of the graph will be thrown out, because we will be summing over the "infinitely thin" rectangles and the height of these "adjacent" rectangles will tell us about the steepness of the graph. On the other hand, I can completely understand why we need the arc length term, because, well, we arn't summing over infinitely thin rectangles, we are summing over the limit as the width of these rectangles goes to zero, and the arc length term tell us about how the ratio of the heights of adjacent rectangles as this limit goes to zero...
Anyway... sometimes I post questions like this here and somebody gives me an insight that blows my mind. Maybe some examples illustrating extreme cases would help...
Thanks in advance!
