Unizor - Definite Integrals - Definition
Channel: Zor Shekhtman on UNIZOR Education
Date: 2017-05-08

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Definite Integrals - Definition

Generalizing theoretical and practical examples presented in the previous lectures, we define
I = lim Σi∈[1,N] f(xi)·Δxi
where f(x) is some smooth real function defined on real segment [a,b],
{xi} is partitioning of segment [a,b] into N parts,
and we assume that N→∞, while the maximum width of intervals Δxi=x[i]−x[i−1] converges to zero
as definite integral of function f(x) on segment [a,b] and will use the following symbolics for it:
I = ∫[a,b]f(x) dx

The above definition belongs to Bernhard Riemann - German mathematician of 19th century and is a definition of Riemann integral.
The sum Σi∈[1,N] f(xi)·Δxi, where Δxi are intervals of partitioning of segment [a,b], is called Riemann sum of function f(x) on segment [a,b].

There are other conceptually different constructions that include the word integral - n-dimensional Riemann integral, curvilinear integral, surface integral, Stieltjes integral, Lebesgue integral, but these concepts are beyond the scope of this course.

As you see, the symbol of definite integral is practically the same as that of indefinite integral. The only difference is the indication of left and right margins of the segment of integration. This is not a coincidence! There is a deep connection between indefinite and definite integrals as will be demonstrated later.

Basic Properties
of Definite (Riemann) Integrals

1. ∫[a,b][f(x)+g(x)] dx =
= ∫[a,b]f(x) dx + ∫[a,b]g(x) dx

This property immediately follows from identity
[f(xi)+g(xi)]·Δxi =
= f(xi)·Δxi + g(xi)·Δxi
and the additive property of the limits.

2. ∫[a,b]f(x) dx = −∫[b,a]f(x) dx

This property follows from identity
(x[i]−x[i−1]) = −(x[i−1]−xi)

3. ∫[a,a]f(x) dx = 0

This follows from the fact that zero-length segment [a,a] produces zero-length intervals [x[i−1],x[i]] by any partitioning.

4. ∫[a,b]c·f(x) dx = c·∫[a,b]f(x) dx

This property follows from the corresponding property of the limit of function multiplied by a constant.

5. ∫abf(x) dx ≤ M·(b−a)
where M is the maximum value of function f(x) on segment [a,b], assuming a ≤ b.

The proof immediately follows from inequality
f(xi)·(xi−xi−1) ≤ M·(xi−xi−1)
after we summarize it by i.

6. ∫abf(x) dx ≥ m·(b−a)
where m is the minimum value of function f(x) on segment [a,b], assuming a ≤ b.

The proof is analogous to a previous one.

7. ∫acf(x) dx =
= ∫abf(x) dx + ∫bcf(x) dx

To prove this, consider a ≤ b ≤ c.
Then each partitioning of segment [a,c] results in partitioning of its two sub-segments [a,b] and [b,c].
The sum of Riemann sums Σi∈[1,N] f(xi)·Δxi on segments [a,b] and [b,c] is the Riemann sum Σi∈[1,N] f(xi)·Δxi on segment [a,c].
Using the additive property of limits, we obtain the final proof.
For all other relative positions