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:
    
 (source: Wayback Machine)
The above links provide interesting applications, like very easy ways to find the area of a cycloid and tractrix.

Question.

*

*What are other new applications?

*Are there new developments of visual calculus?

*What are some similar visual results, which can simplify calculations, and can be included in visual calculus?


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

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