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  18 Integration  Back ] Home ] Next ]    

Chapter 18
Slopes and Areas

Postscript:  Below, Riemann sums are quickly described without writing summation notation.  This summation notation free approach provides tutors and teachers a simpler route for defining the definite integral as limit of Riemann sums. The summation notation is only needed when or if proofs of convergence are required. 

The Second Fundamental Theorem of Calculus says that the area under a curve y = h(x) between endpoints x = a and x = b > a can be expressed in terms of the endpoint values of a function H(x) if

  1. h(x) = H¢(x) = the slope of H(x) when a < x < b, and 
  2. h(x) and H(x) are both continuous on the interval [a,b].
This chapter indicates why the second fundamental theorem holds. The solution of an area-under-a-curve calculation problem implies the conclusion of this theorem.

Areas Definition: Covering by Squares or Rectangles


The area of the region A under the curve y = h(x) between x = a and x = b could be approximated by covering the region by squares, all of the same size, and then allowing the size (measured by their width) approach zero. The limiting value of the approximations should be a finite value. It will be if the curve y = h(x) is continuous (Proof Omitted). The limiting value is declared to be the area of the region. That is a mathematical definition.

The squares contributing to these estimates (counted in) may be grouped together into rectangles, horizontally or vertically say. This provides a connection to the alternate area approximation based on thin, horizontal or vertical rectangles in calculus.
The fifth chapter in the text Calculus by Lipman Bers (Holt, Rinehart and Winston 1969, SBN 03-065240-5) makes this connection as well.
See below.
An alternate mathematical definition is to cover the region by n rectangles, all of the same width, with one end on the horizontal axis, and the other end on the curve. For the area drawn, a covering by nine such rectangles is shown below. The height of the rectangles shown equals the value of the function at the midpoint of the base.

Adding all the areas of the rectangles together yields a Riemann sum. 

The total area covered by the rectangles is given by a Riemann Sum. As the number n of equi-width rectangles increases, the common width and base size tends to zero and the Riemann sums should tend to a limiting value. This limiting value is called the area under the curve at least when h(x) is nonnegative between x = a and x = b.

The First Fundamental Theorem of Calculus says that if h(x) is continuous between and at the endpoints x = a and x = b, that is continuous on the closed interval [a,b], then all the rectangle-based approximations approach a single finite limiting value as the width of the base tends to zero. This limiting value of the sum of rectangle areas, a Riemann sum, provides the computational definition of the area-under-a-curve in calculus.

FOOTNOTE: Approximation of the area under a continuous curve h(x) ³ 0 by small squares yields the same value in the limit. For each square with one size on the horizontal axis, the union of it with the squares above it form a rectangle with one end on the horizontal axis and another end on the curve, or very close to the curve.

A proof of the First Fundamental Theorem of Calculus is given in the appendices.     
Optional:: Riemann sums may be written in summation notation as
n
å
j = 1 
h(xj)Dx = h(x1)Dx + h(x2)Dx + ¼+h(xn)Dx
where xj represents a point on the base of the jth rectangle. Such points can be picked at random or not. Non-random choices are provided by midpoints, left-endpoints, right-endpoints or the location of the greatest value of h(x) or the least in each rectangle base. Each base has length |Dx| = [(|b-a|)/(n)].

Our initial concept or intuition of the area under a curve requires that h(x) be nonnegative, but the First Fundamental Theorem of Calculus does not require this. The term h(xj)Dx represents the signed area of a rectangle with base of width |Dx| on the horizontal x-axis and another end on the horizontal line y = h(xj). The signed area is negative when h(xj) < 0 and Dx > 0. The limit of the Riemann sum, area approximations yields a signed area between the curve and the horizontal x-axis. The intervals where the function h(x) is positive make a positive contribution to the signed area. The intervals where the function h(x) is negative make a negative contribution to the signed area when a < b.


Why Slopes
and
More Math

understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7

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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc


If  you like Volume 3  you may also like  Three Skills for Algebra , Exponents & Radicals Exactly,  complex numbers, Euclidean Geometry , More Calculus and  Pattern Based Reason  as well. 

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity


These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem
is a postscript,
not in printed version.


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