Riemann Stieltjes Integral help

  • Context: Graduate 
  • Thread starter Thread starter prudens2010
  • Start date Start date
  • Tags Tags
    Integral Riemann
Click For Summary
SUMMARY

The discussion focuses on calculating the Riemann-Stieltjes integral of the function f defined as f(x) = x for 0 ≤ x ≤ 1 and f(x) = 1 for x > 1, with respect to the function α defined as α(x) = x² for 0 ≤ x ≤ 1 and α(x) = 1 for x > 1. The integral is evaluated from 0 to 23, simplifying to an ordinary Riemann integral due to the piecewise nature of both functions. The differential dα is determined as dα = 2xdx for 0 ≤ x ≤ 1 and dα = 0 for x > 1, leading to the conclusion that the integral can be computed directly using standard integration techniques.

PREREQUISITES
  • Understanding of Riemann-Stieltjes integrals
  • Familiarity with piecewise functions
  • Knowledge of basic integration techniques
  • Concept of differentials in calculus
NEXT STEPS
  • Study the properties of Riemann-Stieltjes integrals
  • Learn how to evaluate piecewise functions in integrals
  • Explore advanced integration techniques in calculus
  • Investigate applications of Riemann-Stieltjes integrals in real analysis
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus and real analysis, as well as educators looking for examples of Riemann-Stieltjes integrals.

prudens2010
Messages
1
Reaction score
0
Code: 
Let f = 

x, for 0<=x<=1
1, for 1<x

alpha =

x^2, for 0<=x<=1
1, for 1<x

Find Integral (f) d(alpha) -- from 0 to 23

pls help!
 
Physics news on Phys.org
prudens2010 said:
Code: 
Let f = 

x, for 0<=x<=1
1, for 1<x

alpha =

x^2, for 0<=x<=1
1, for 1<x

Find Integral (f) d(alpha) -- from 0 to 23

pls help!

Use the definition of the R-S integral. The answer should pop right out unless I'm missing something. Where does the problem lose you?
 
This case easily translates into an ordinary Riemann integral.
dα = 2xdx, 0≤x≤1
dα = 0, x>1
 

Similar threads

Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad Relating integration of forms to Riemann integration
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Challenging integral involving exponentials and logarithms
  • · Replies 14 ·
Replies
14
Views
3K
MHB How to prove this corollary in Line Integral using Riemann integral
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Another Integration Formula
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Calculating the change of the volume of a sphere using this integral
  • · Replies 3 ·
Replies
3
Views
3K
High School Arc Length for Hyperbolic Sin
  • · Replies 6 ·
Replies
6
Views
3K
High School Riemann integrability of functions with countably infinitely many dis-
  • · Replies 3 ·
Replies
3
Views
3K
MHB Interval of the Riemann integral value
  • · Replies 3 ·
Replies
3
Views
2K