Integrator monotonically increasing ? R.S. Integral

In summary, the R.S. Integral is a type of integral similar to the Riemann integral, but instead of using the size of each interval, it uses the difference in values of a monotone increasing function. This allows for integration with functions that are not differentiable, such as the unit step function. If \alpha is a differentiable function, the Riemann-Stieljes integral is equivalent to the Riemann integral. However, if \alpha is not differentiable, the R.S. Integral can still be used to find the integral.
  • #1
sihag
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Could someone explain why the R.S. Integral is defined for a monotonically increasing integrator? Can't we use a decreasing fuction anologously?
 
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  • #2
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].)
 

What is an Integrator monotonically increasing?

An Integrator monotonically increasing is a mathematical method used to calculate the area under a curve, also known as the integral, by adding up smaller segments of the curve. It is called "monotonically increasing" because the value of the integral increases as the curve moves to the right, and does not decrease or fluctuate.

How is an Integrator monotonically increasing used in science?

An Integrator monotonically increasing is commonly used in science, particularly in physics and engineering, to calculate quantities such as velocity, acceleration, and energy. It is also used in data analysis to find the average value of a continuous function.

What is the difference between an Integrator monotonically increasing and a Riemann sum?

An Integrator monotonically increasing is a more accurate and precise method of calculating integrals compared to a Riemann sum. While a Riemann sum uses rectangles to approximate the area under a curve, an Integrator monotonically increasing uses smaller and more accurate segments, resulting in a more precise value for the integral.

Are there any limitations to using an Integrator monotonically increasing?

One limitation of using an Integrator monotonically increasing is that it requires the function to be continuous. Discontinuous or discontinuous functions may result in inaccurate values for the integral. Additionally, this method may be computationally intensive and time-consuming for more complex functions.

What is the significance of the R.S. Integral in an Integrator monotonically increasing?

The R.S. Integral, also known as the Riemann-Stieltjes integral, is a more general form of the Integrator monotonically increasing that allows for the integration of functions with discontinuities or jumps. It is used when the function being integrated is not continuous, making it a more versatile method compared to the standard Integrator monotonically increasing.

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