Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integrator monotonically increasing ? R.S. Integral

  1. Feb 20, 2008 #1
    Could someone explain why the R.S. Integral is defined for a monotonically increasing integrator? Can't we use a decreasing fuction anologously?
  2. jcsd
  3. Feb 20, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you could. It would just give the negative of the corresponding integral with an increasing integrator.

    (For anyone who is wondering, "R.S" is the Riemann-Stieljes integral. It is defined exactly like the Riemann integral except that instead of measuring the size of each interval forming the base of a rectangle as [itex]x_{i+1}- x_i[/itex], we use [itex]\alpha(x_{i+1})-\alpha(x_i)[/itex] where [itex]\alpha(x)[/itex] can be an monotone increasing function of x. It is typically written [itex]\int f(x)d\alpha(x)[/itex]. If [itex]\alpha[/itex] is a differentiable function, the Riemann-Stieljes integral is exactly the same as the Riemann integral [itex]\int f(x) d\alpha/dx dx[/itex]. The interesting situation is when [itex]\alpha[/itex] is not differentiable. In particular, if [itex]\alpha[/itex] is the unit step function, and a and b are integers, then [itex]\int_a^b f(x) d\alpha= f(a)+ f(a+1)+ \cdot\cdot\cdot+ f(b)[/itex].)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Integrator monotonically increasing ? R.S. Integral
  1. On Integration (Replies: 4)

  2. An integral (Replies: 2)