Abstract
In this chapter we present the concept of volume of a solid and compute the volumes of various of the solids studied so far. Cavalieri’s principle will turn to be a central tool for our exposition; in particular, we shall use it to compute the volume of a sphere of a given radius.
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- 1.
Bonaventura Cavalieri , Italian mathematician of the sixteenth century.
- 2.
The cornerstone of the usual notion of measurability of solids is the concept of Jordan content (cf. [19] or [22]); in this context, one proves that a solid is measurable if and only if its boundary has null Jordan content. In the modern theory of measure and integration, it is possible to show (cf. Chapter 2 of [9], for instance) that our definition of measurability is implied by the usual one, and that the two versions of Cavalieri’s principle we present here, as well as our Postulates 2., 3. and 5. for the measurement of volumes (see below), are theorems.
- 3.
What must be proved is that the intersection of the solid with any plane that intersects its interior is a region of measurable area. If the plane is transversal to the surface of revolution that defines the solid (in the sense that the plane is not tangent to the surface at any point), then the measurability of the corresponding planar section is a fairly simple consequence of the Implicit Function Theorem (see [22]). The general case follows from this one, through a slightly more complicated argument.
References
T. Apostol, Calculus, Vol. 2 (Wiley, New York, 1967)
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
G.B. Folland, Real Analysis: Modern Techniques and Their Applications (Wiley, New York, 1999)
E.E. Moise, Elementary Geometry from an Advanced Standpoint (Addison-Wesley, Boston, 1963)
W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, New York, 1976)
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Caminha Muniz Neto, A. (2018). Volume of Solids. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_13
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