1—Basic Stuff 12work in this book, but it is a fairly simple extension of the Riemann integral and I’m introducing it
mostly for its cultural value — to show you that thereareother ways to define an integral. If you take
the time to understand it, you will be able to look back at some subjects that you already know and to
realize that they can be manipulated in a more compact form (e.g. center of mass).
When you try to evaluate the moment of inertia you are doing the integral
∫
r^2 dm
When you evaluate the position of the center of mass even in one dimension the integral is
1M
∫
xdm
and even though you may not yet have encountered this, the electric dipole moment is
∫~rdq
How do you integratexwith respect tom? What exactly are you doing? A possible answer is that
you can express this integral in terms of the linear density function and thendm=λ(x)dx. But if the
masses are a mixture of continuous densities and point masses, this starts to become awkward. Is there
a better way?
Yes
On the intervala≤x≤bassume there aretwofunctions,fandα. Don’t assume that either of them
must be continuous, though they can’t be too badly behaved or nothing will converge. This starts the
same way the Riemann integral does: partition the interval into a finite number (N) of sub-intervals
at the points
a=x 0 < x 1 < x 2 < ... < xN=b (1.27)
Form the sum
∑Nk=f(x′k)∆αk, where xk− 1 ≤x′k≤xk and ∆αk=α(xk)−α(xk− 1 ) (1.28)
To improve the sum, keep adding more and more points to the partition so that in the limit all the
intervalsxk−xk− 1 → 0. This limit is called the Riemann-Stieltjes integral,
∫
f dα (1.29)
What’s the big deal? Doesn’tdα=α′dx? Use that and you have just the ordinary integral
∫
f(x)α′(x)dx?
Sometimes you can, but what ifαisn’t differentiable? Suppose that it has a step or several steps? The
derivative isn’t defined, but this Riemann-Stieltjes integral still makes perfectly good sense.