Help with Proof on Integration

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    Integration Proof
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Homework Help Overview

The problem involves proving the integrability of a function g(x) that differs from a bounded and integrable function f(x) on only finitely many points. The task is to demonstrate that g(x) is integrable and that the integrals of f(x) and g(x) over the same interval are equal.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the Fundamental Theorem of Calculus as a starting point and explore the implications of modifying a function at finitely many points. Questions about the definition of integrability and the area interpretation of integrals are raised.

Discussion Status

Participants are actively engaging with the problem, offering tips and prompting further exploration of definitions and concepts related to integrability. There is an emphasis on understanding the implications of removing points from a function's graph.

Contextual Notes

There is a focus on the area interpretation of integrals and the properties of functions with respect to points of discontinuity. The discussion is framed within the context of a classroom assignment, and the original poster expresses uncertainty about the theorem's identification.

tomhawk24
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I was assigned this problem in class. My instructor said it was a very popular theorem, but I cannot find it in my book or online. I am clueless on what to do. I would appreciate the help.

Let f(x) be bounded and integrable on [a, b]. Assume that g(x) differs from f(x) on only finitely many points in the domain. Show that g(x) is integrable. Moreover, show that ∫f(x)dx = ∫g(x)dx (Both integrals are from b to a).
 
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welcome to pf!

hi tomhawk24! welcome to pf! :smile:

start with the definition …

which definition of integral (or integrable) are you using?
 
Well we are working mainly on the Fundamental Theorem of Calculus right now.
 
ok, we'll start with that, then …

what does the fundamental theorem of calculus say? :smile:​
 
Starting from the area interpretation of the integral, answer this question: If I take finitely many points out of the graph of a curve f(x) and place them at some other y-coordinate, would the function still be integrable? What would be its integral?

Tip: Does a point have dimensions, or does a line have width? What is the area of a rectangle?
 
First show
∫(f(x)-g(x))dx =0
then use linearity
 

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