Unizor - Definite Integrals - Area under Curve
Channel: Zor Shekhtman on UNIZOR Education
Date: 2017-04-28

Unizor - Creative Minds through Art of Mathematics - Math4Teens

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

Definite Integrals - Area under Curve

Consider the following problem.
Given a smooth function f(x) (we will always consider smooth functions in terms of continuity and sufficient differentiability), with non-negative values (that is, f(x) ≥ 0), defined on a closed segment [a, b] and represented by its graph on (X,Y) coordinate plane.

Our task is to find the "area under curve" - the measure of a part of coordinate plane bounded on the top by a graph of this function, on the bottom - the X-axis, on the left - by a line x = a and on the right - by a line x = b.

There is no ready to use formula for such an area. We do know how the area of a rectangle is defined, it is a product of its two dimensions - length multiplied by width, but not of such a complicated figure as the one we consider now.
We really have to define what the area of this figure is and then attempt to calculate it based on values of function f(x) on segment [a, b].

The process of approximation starts with dividing segment [a, b] into N intervals by points a=x0, x1, x2, x3...xN=b, and constructing rectangles, based on each interval [xi−1, xi], having the height equaled to a minimum value mi of function f(x) on this interval.
The sum of their areas does not exceed the intuitively understood "area under curve".

Alternatively, we can consider a set of rectangles built on the same intervals but of the height equaled to a maximum value Mi of function f(x) on each interval (consider now pink extension of light blue rectangles).
New taller rectangles are higher than the curve and, therefore, the sum of their areas is not less than the intuitively understood "area under curve".

sN = Σi∈[1,N]mi·(xi−xi−1) =
= Σi∈[1,N]mi·Δxi−1
where Δxi−1 = xi−xi−1, which is the width of each interval.

SN = Σi∈[1,N]Mi·(xi−xi−1) =
= Σi∈[1,N]Mi·Δxi−1
where Δxi−1 = xi−xi−1, which is the width of each interval.

Both approximations seem to be better if the number of intervals N we divide our segment [a, b] is large. And the larger is N - the better our approximation will be.

This can be confirmed by the following obvious statements.
If we divide any existing interval in two parts, build two rectangles instead of one and calculate the sum of areas of these rectangles, sum sN, based on minimum values of function f(x), will increase and sum SN, based on maximum values of function f(x), will decrease.

In this lecture we prove that both sN and SN have limits and these limits are the same, thus allowing to define the area under curve as this limit.