Applications of visual calculus

Mamikon's visual calculus (see Mamikon, Tom Apostol, Wikipedia) is a very beautiful and surprisingly efficient tool.

The basis is

Mamikon's theorem. The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.

For a nice picture, see this, and the following picture from Apostol's introduction:

The above links provide interesting applications, like very easy ways to find the area of a cycloid and tractrix.

Question.

1. What are other new applications?
2. Are there new developments of visual calculus?
3. What are some similar visual results, which can simplify calculations, and can be included in visual calculus?
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This doesn't answer your question, but Mamikon's theorem leads to an appealing solution to the following (popular) calculus problem, while the standard solution is rather brute force: media.cheggcdn.com/media/477/… –  Dustin G. Mixon May 12 '13 at 15:54
@Dustin G. Mixon: that's a cool application, and I think it can be made into an answer. –  Cristi Stoica May 12 '13 at 18:54
@Cristi Stoica: Done. –  Dustin G. Mixon May 12 '13 at 19:32
I honestly didn't think I'd ever learn anything about calculus this new to me at this stage in my education. –  Ryan Reich May 13 '13 at 3:26
very nice result! That would bring some 'fun' back to mathematical education, but I almost fear that reasons for not presenting it in the class room will be the same as the one's for not anticipating it back in 1959. –  Manfred Weis Feb 26 '14 at 6:14

Mamikon's theorem leads to an appealing solution to the following (popular) calculus problem, while the standard solution is rather brute force:

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Perhaps this previous MO question may help: Taking “Zooming in on a point of a graph” seriously, e.g., this answer link.

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I'm not sure that the following is what you are looking for but I hope that it sheds some useful light on the topic of your query and suggests further applications. Given a curve in the plane with parametrisation $(c_1(u),c_2(u))$ one can consider the transformation $$F(u,v)=(c_1(u)+\sqrt {2 v} \dot c_1(u),c_2(u)+\sqrt {2 v}\dot c_2(u)).$$ (We are actually interested in the network this mapping introduces in the plane---the image of the coordinate network---which has a natural geometrical interpretation related to the OP). A simple computation shows that the determinant of the Jacobi matrix of this mapping is $\ddot c_1(u)\dot c_2(u)-\dot c_1(u)\ddot c_2(u))$. From this we can deduce various useful facts:

$1.$ The parametrisation $c$ does not appear explicitly (only its derivatives). This is the reason for the satement at the start of the OP.

$2.$ The determinant is independent of $v$ (this was the reason for the strange dependence of $F$ on $v$). In particular we can choose a parametrisation for $c$ for which this is identically $1$ which means that $F$ is area-preserving. This can be used to garner a plethora of results for particular curves.

$3.$ The case of the cycloid has some special features which explains some results and methods in the works quoted If we use the standard parametrisation $(t-\sin t,1-\cos t)$, then the above determinant is $1-\cos t$ which is just the height of the given point above the $x$-axis.

Much more could be said about this topic, but we would like to close with the remark that these facts were not just pulled out of thin air---behind them there lies an important concept, that of a Samuelson configuration, which was introduced by the economics laureate Paul Samuelson (not under that name, of course) in his Nobel acceptance speech, i.e., over 40 years ago.

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