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We know that for a cylinder $$ V = \pi r^2 h $$ This formula is easily visualized as a stack of $h$ circles with radius $r$.

However, as a little experiment with the goal of trying to think about things differently, I attempted to do the same, but using rectangles instead. Naturally, at least to me, I visualized a circular cylinder as rectangles, with width $r$ and height $h$, revolved around the center point of the cylinder.

I thought that one could simply then say that the volume should be the area of the rectangle * the circumference of the cylinder with this calculation. $$ V = rh \cdot 2\pi r = 2\pi r^2 h $$ Obviously, this seems to not be true and is what is getting to me. Can anyone explain why this is not true? What am I missing here?

To further clarify, let's simplify it so we can visualize it with something physical. Take your phone, or anything rectangular. Now rotate it 180 degrees. We just made a cylinder by rotating a rectangle, with width 2r and length h, (pi)r times. $$ V = 2rh * \pi r = 2\pi r^2 h$$

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  • $\begingroup$ There's a related question here which might help you see your error. $\endgroup$ – 6005 Oct 4 '16 at 14:45
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When you stack up the circles, each stack has length $h$. When you revolve the rectangle, the distance a point travels increases with its radius from the center. If a point is at distance $x$ from the center, it will travel $2 \pi x$ in the revolution, so the volume is $$V=\int_0^r 2 \pi x h \; dx=\pi r^2h$$

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  • $\begingroup$ I can see it this way, but when looking at it intuitively, I'm still bumfuzzled as to why it seems that the number of rectangles that make up the cylinder isn't $$ 2 \pi r $$ $\endgroup$ – grundyboy34 Oct 4 '16 at 15:23
  • $\begingroup$ I would say there are that many, but each one does not make the volume you think because it doesn't sweep as far near the center. $\endgroup$ – Ross Millikan Oct 4 '16 at 15:57
  • $\begingroup$ Ahh I see now. The number of rectangles changes as we travel across r , since the circumference will change. $\endgroup$ – grundyboy34 Oct 4 '16 at 16:04
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When you multiply the area by the circumference it means that you are multiplying the area by the far most point of revolution (which is wrong). The correct way is to multiply the area by the average of circumference which is $$\frac {2 \pi r}{2}={\pi} {r}$$ Therefore $V = rh \cdot \pi r=$$ \pi r^2h$

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