Additive Property of Riemann Integrals | Real Analysis | B.Sc./B.A. (Mathematics) Third Year, TU

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Kshitiz Mangal Bajracharya

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2,500

Published on October 8, 2024 12:00:00 AM ● Video Link: https://www.youtube.com/watch?v=u6R8Crywh4I

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In this video, we discuss the statement and proof of the additive property of Riemann Integrals which states that the sum of two Riemann integrable functions is Riemann integrable and the integral of the sum of the two functions is equal to the sum of integrals of the given function.

Time Stamps.

00:00 Intro

00:16 Statement of the additive property

00:55 Proof starts

01:05 Result on Upper & Lower sums related to sum of two functions required for this proof

02:31 Two parts of the proof - approach outlined

03:10 f+g is Riemann integrable ... proof begins

03:51 Step 1 - Using Riemann's condition on given functions

07:31 Step 2 - Defining a new partition of [a, b] for using it to prove that f+g is Riemann integrable

09:39 How do upper and lower sums change when a partition is refined?

11:29 Step 3 - Proving Riemann's condition for f+g, making it Riemann integrable

17:29 Integral of (f+g) = integral of (f) + integral (g) ... proof begins

18:29 Concepts to be used in this part of the proof

18:59 Reconsidering Riemann's condition and defining same partitions as before

20:51 Step 1 - Integral of (f+g) is less or equal to integral of (f) + integral (g)

21:12 Using result on upper sum related to sum of two functions in step 1

21:27 Using infimum and upper Riemann Integral in step 1

22:41 Using the fact that upper sum decreases or remains the same when the partition is refined, in step 1

23:03 Using Riemann's condition in step 1

25:16 Using supremum and lower Riemann integral in step 1

25:51 Using Riemann integrability (given conditions and first part of the proof) and completing step 1

26:31 Step 2 - Integral of (f+g) is greater or equal to integral of (f) + integral (g)

27:00 Using result on lower sum related to sum of two functions in step 2

27:15 Using supremum and lower Riemann Integral in step 2

28:31 Using the fact that lower sum increases or remains the same when the partition is refined, in step 2

29:20 Using Riemann's condition in step 2

30:25 Using infimum and upper Riemann integral in step 2

31:00 Using Riemann integrability (given conditions and first part of the proof) and completing step 2

32:29 Proof ends

33:00 Outro

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