Three results from three mathematicians from three different eras provide interesting and surprisingly easy instruments to compute areas and volumes of plane figures or solids.
The first two - Cavalieri's Principle (1635) and Mamikon's Theorem (1959) - directly supply stunningly calculus-free methods for calculating "integrals" (possibly by hiding calculus under the carpet). At first glance, they seem incredibly similar in fashion; yet, I don't believe that they are quickly (and calculus-freely) deducible one from the other.
The third one is a not-so-well-known proposition by Italian mathematician Giuseppe Peano, who discussed it in his "Applicazioni geometriche del calcolo infinitesimale" (1887). It is much more general and it implies both Cavalieri's result for areas and Mamikon's Theorem as corollaries. Unluckily, it's calcululus-heavy both in its statement and its proof.
I will summarise the three statements (for the sake of simplicity, just very stripped-down 2D-versions) in a short while. First of all, though, I want to make my question explicit: is there a way to formulate a proposition which is general enough to include both Cavalieri's Principle and Mamikon's Theorem, and at the same time "narrow" enough not to require calculus at least in its statement, as Peano's result does instead?
Cavalieri's Principle (Wikipedia):
Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.Mamikon's Theorem (Wikipedia: Visual Calculus):
The area of a tangent sweep to a curve is equal to the area of its tangent cluster, regardless of the shape of the original curve. (Tangent sweep: the surface "swept" by a family of segments tangent to a curve when one of their endpoints P continuously moves along the curve; tangent cluster: the surface obtained by translating the aforementioned segments so that P is no longer moving - see figure)
Peano's result (Archive):
Let $A(t), B(t)$ be plane $C^1$ curves, parametrised by $t$, such that the segment $AB(t)$ never passes through the same point as $t$ ranges from $t_0$ to $t_1$. The area swept by the moving segment $AB(t)$ is given by:
$\int_{t_0}^{t_1}(B(t)-A(t))\cdot(\frac{d}{dt} A(t)+\frac{d}{dt} B(t))dt$.
Special cases of Peano's result give Cavalieri's and Mamikon's ones, as discussed here by Gabriele Greco et al.:
Particular instances of the formula considered by Peano are the following:
a) The point A moves along a straight line and the angle of the segment AB with that line is constant;
b) The point A is fixed;
c) The segment AB is tangent at the point A to the curve described by A;
d) The segment AB is of constant length and normal to the curve described by its midpoint.
Case a) brings Cavalieri's Principle; case c) (perhaps in combination with b)) gives Mamikon's Theorem.
So, question is: is it possible to "merge" Cavalieri and Mamikon (+ possibly other corollaries of Peano) without any direct reference to the integral? Something more in the style of Cavalieri and Mamikon, such as: "if line segments constructed in {some general way encompassing both Cavalieri's and Mamikon constructions} in both figures are the same, then the two figures are area-equivalent"? I don't actually care about avoiding integrals in the proof: it's satisfying enough for me if the theorem statement is integral-free.
And what about volumes?