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 = = = = 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 = .

The slice has dA = dx.

The area of the top half of the circle is given by the
following integral:

whose solution Pi a^{2} /2 can be found using the
trigonometric substitution x = a sin(t).

The ellipse is defined by

+ = 1

Multiplying by a^2, and then solving for y gives for the top half

y =

Thus a slice of an ellipse has dA = 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:

= Pi a^{2}
/2 = Pi a b /2

In summary, we found the area of a circle (Pi a^{2} ) 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.