Scaling and Slicing

Acknowledgements: Much of the following material is adapted from or inspired by The Pleasures of Counting by T.W. Körner, Cambridge University Press, 1996 and Calculus&Mathematica by Bill Davis, Horacio Porta, and Jerry Uhl, Addison-Wesley Publishing Company, 1994.

1. Slicing.

The idea of "slicing" is one of the most fundamental and powerful ideas of integral calculus. The basic concept is contained in the definition of the definite integral as a Riemann sum. In this case the hard problem of finding the area under an arbitrary curve is attacked by slicing the area into rectangles whose area is easily determined. In the limit of adding up infinitely many infinitely-thin slices we get an integral which represents a "formal" solution to the original problem.

Finding the area under a curve is not the only problem that can be solved by slicing. The same approach is used for finding the volume of a solid of revolution by the disk method, for finding the hydrostatic force on a submerged object, for finding the moments of a system about the x and y axes, for finding the arc length, for finding the surface area of a solid of revolution, etc.

2. Scaling.

The relationships between quantities that I call "scaling" are actually treated in a more formal way in dimensional analysis. The goal here is to take some simple scaling relationships and show that they can be used to extend the power of "slicing" (following Davis, Porta, and Uhl).

If an object is scaled up or down in size in a linear fashion so that any length changes by the same (constant) scale factor, then the following well known result holds true: Pick some characteristic length, L. The area will scale as L^2, and the volume will scale as L^3. This simple relationship has huge implications for biological and physical systems. For example, the strength of a bone or column is proportional to its cross sectional area, L^2, but the weight of the animal or structure is proportional to its volume, L^3. This puts a major constraint on the shape and size of animals or structures. For many more interesting examples, see Körner where using scaling arguments he shows that fleas, humans, kangaroos, and grasshoppers all jump approximately the same height!

3. The paradox of Gabriel's Horn.

Gabriel's Horn is the surface generated by rotating 1/x about the x axis for x > 1.



You can fill it with paint, but you can't paint it!

One possible explanation of the paradox is with a scaling argument. As x increases the horn is shaped like a very long thin needle. Pick the diameter of the needle, D, as the characteristic length scale. As x goes to infinity, D = 1/x goes to 0. The surface area ~D^2, but the volume ~D^3. As D goes to 0, the surface area divided by the volume ~1/D which goes to infinity. Hence the volume of the thin needle approaches 0 faster than the surface area as x goes to infinity. Thus it is at least plausible that the surface area is infinite while the volume is finite. (Note that there is no problem with an infinite object being finite in some respect. For example, you could cut up a pie in pieces of 1/2, 1/4, 1/8, ... which add up to exactly one whole pie even though there are an infinite number of pieces.)


4. Cones.

Consider a cone that has height, h, and base of area B. Let s be the distance measured downward from the top of the cone. Furthermore, let s be the characteristic length. Then from our scaling arguments the area of a slice made parallel to the base is A(s) = k s^2, where k is some constant. Note that when s = h we have A(h) = B = k h^2. Thus k = B / h^2. Now we know that the area of a horizontal slice is A(s) = k s^2 = B s^2 / h^2.


The slice has volume dV = A(s) ds. Hence the volume of the cone is

V = [Graphics:Slicinggr10.gif] = [Graphics:Slicinggr11.gif] = [Graphics:Slicinggr12.gif] = B h / 3

Note that nothing has been said about the shape of the base of the cone. It need only have area B. Furthermore, nothing in the analysis above required that the cone be vertical. In other words, the critical scaling argument A(s) = k s^2 holds if s is taken vertically even if the cone itself is not vertical. This means that the slices can be shifted horizontally in any manner whatsoever, as long as the horizontal slice does not change its shape or size. Thus the following cones all have the same volume:


5. Ellipses and Circles.

Circles and ellipses are symmetric about the x axis in this example, so in the following we deal with the top half for convenience. The equation for a circle of radius a is x^2 + y^2 = a^2. Solving for y and taking the positive square root gives the equation for the top half of the circle, y = [Graphics:Slicinggr14.gif].


The slice has dA = [Graphics:Slicinggr16.gif] dx.

The area of the top half of the circle is given by the following integral:
whose solution Pi a2 /2 can be found using the trigonometric substitution x = a sin(t).
The ellipse is defined by
[Graphics:Slicinggr19.gif] + [Graphics:Slicinggr20.gif] = 1
Multiplying by a^2, and then solving for y gives for the top half
y = [Graphics:Slicinggr21.gif][Graphics:Slicinggr22.gif]


Thus a slice of an ellipse has dA = [Graphics:Slicinggr24.gif][Graphics:Slicinggr25.gif]dx.

Each slice of the ellipse is the same as a slice of the circle scaled by a factor of b/a.


Thus the area of the top half of the ellipse is the area of the top half of the circle multiplied by b/a:
[Graphics:Slicinggr27.gif] = [Graphics:Slicinggr28.gif] Pi a2 /2 = Pi a b /2
In summary, we found the area of a circle (Pi a2 ) by traditional means, and then used scaling arguments to find the area of an ellipse (Pi a b) without evaluating any integrals.


6. Conclusion.

Scaling arguments have been used to help explain the results of certain improper integrals (Gabriel's Horn). Simple scaling arguments (from dimensional analysis) can tell us a great deal about biological or physical systems without knowing the formal functionality governing a particular process.
Scaling arguments can be combined with the concept of slicing from integral calculus to extend the power of slicing. This was illustrated with two examples: the volume of cones and the area of ellipses.