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
- h(x) = H¢(x) = the slope of H(x) when a < x < b, and
- h(x) and H(x) are both continuous on the interval [a,b].
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.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.
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.
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.
A proof of the First Fundamental Theorem of Calculus is given in the appendices.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.
Optional:: Riemann sums may be written in summation notation as
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
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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:
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
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.