Help with Proof on Integration

[tomhawk24]

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).

 

[tiny-tim]

welcome to pf!

hi tomhawk24! welcome to pf!

start with the definition …

which definition of integral (or integrable) are you using?

 

[tomhawk24]

Well we are working mainly on the Fundamental Theorem of Calculus right now.

 

[tiny-tim]

ok, we'll start with that, then …

what does the fundamental theorem of calculus say? ​

 

[Millennial]

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?

 

[lurflurf]

First show
∫(f(x)-g(x))dx =0
then use linearity