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Chapter 18
Slopes and Areas

Volume 3, Why Slopes and More Math.

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].

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, those counted as 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.

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.

Approximation of the area under a continuous curve $h(x)$ > 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.

Area Calculation Problem

The selection of labels x and y for the horizontal and vertical axes for most of graphs met so far is arbitrary. It can be changed. The letters s and q could have been used instead in all the previous graphs. You should imagine this replacement, and the effect, if any, it has on your knowledge or opinion of mathematics.

Problem: Suppose $G(x)$ and $h(x)$ are continuous at each point $x$ in the interval $[a,b]$. Further suppose that the slope $G'(x)$ of $G(x)$ satisfies $G'(x)=h(x)$ for $a\le x \le b$. \bf Find a formula for the area $A$ under the curve $q=h(s)$

Step 1. Define an Area Function

First, introduce a function F(x) as follows. For each x between a and b inclusive let

Here $F(a)=0$ and $F(b)=A$ is the area to be computed. Note that the area computation will be based on (a) finding the slope or derivative $F'(x)$ and then (b) observing how to obtain $F(x)$ from a knowledge of $F'(x)$. The value of $A=F(b)$ is required. It will be given by a formula involving the function $G(x)$.

Step 2. Area Function Slope Calculation

Second, the following diagram leads to a formula for $F'(x)$.

In this figure, the area under $q=h(s)$ from $s=x$ to $s=x+\Delta x$ is given by \[ \Delta F=F(x+\Delta x)-F(x) \approx h(x)\Delta x \] and hence \[ \frac{F(x+\Delta x)-F(x)}{\Delta x} \approx h(x) \] These approximations are expected to improve when $\Delta x \to 0$ approaches zero. This hope or expectation suggests that \begin{eqnarray*} F'(x)&=&\lim_{\Delta x \to 0} \frac{\Delta F}{\Delta x} \\ &=&\lim_{\Delta x \to 0} \frac{ F(x+\Delta x)-F(x)}{\Delta x} \\ &=& h(x) \end{eqnarray*} Therefore the derivative or slope function for $F(x)$ should be\[ F'(x)= h(x) \] This formula gives the rate of change of area $F(x)$ for each given value of $x$. The small print below provides a more refined argument to justify the assertion that $F'(x)=h(x)$.

Since $h(s)$ is assumed to be continuous on the interval $[a,b]$, it is continuous at the point $s_1=x$. Therefore, for every whole number $k$, there is a number $n$ such that \[|h(s)-h(x)| \le \frac12\cdot 10^{-k} \] when $|s-x| \le \frac12\cdot 10^{-n}$. Now given $k$ and such an $n$, if $x \le s \le x+\Delta x$ and $ 0 \le \Delta x \le \frac12\cdot 10^{-n}$ then $|s-x| \le |\Delta x| \le \frac12\cdot 10^{-n}$ as well. For such numbers $s$, it follows that $|h(s)-h(x)| \le \frac12\cdot 10^{-k}$. The latter in turn - see diagram - implies \rm that the region $B$ [between $q=h(x)$ and $q=h(s)$ above the $s$-interval from $s=x$ to $s=x+\Delta x$] has an area

\begin{eqnarray*} |\mbox{Area B}|&=&\left|F(x+\Delta x) -F(x) -h(x)\Delta x\right| \\ &\le & \Delta x \cdot \frac12\cdot 10^{-k} $

This is equivalent to

$\left|\frac{\mbox{Area B}}{\Delta x}\right|=\left|\frac{ F(x+\Delta x) -F(x) -h(x)\Delta x}{\Delta x}\right| \le \frac12\cdot 10^{-k} $

Therefore

$\left|\frac{\mbox{ Area B}}{\Delta x}\right|= \left|\frac{F(x+\Delta x) -F(x)}{\Delta x} -h(x)\right| \le \frac12\cdot 10^{-k} $

when $0 \le \Delta x \le \frac12\cdot 10^{-n}$. Since the foregoing argument holds for every whole number $k>0$, it implies that the limiting value of $\frac{F(x+\Delta x) -F(x)}{\Delta x} =h(x)$ when $\Delta x >0$ approaches zero.

Step 3. Difference of Two Functions

Third, the previous discussion of vertical motions (and earthquakes) implies or suggests that for $x$ in the interval $[a,b]$, the difference $F(x)-G(x)=C$ for some constant $C$ which does not depend on $x$.

Recall F(a)=0. This implies \[C=F(a)-G(a)=0-G(a)=-G(a)\] Therefore, the constant value is given by $C=-G(a)$ \bf because \rm F(a)=0. Now F(x)-G(x) = C implies \[F(x)=G(x)+C=G(x)+(-G(a))=G(x)-G(a)\] Therefore $F(x)=G(x)-G(a)$ and the sought-after area \[A=F(b) = G(b)-G(a)\]

Remark. The latter formula is correct whenever $G(x)$ is a function whose slope or derivative is $h(x)$ for every $x$ in the interval $[a,b]$. Given a formula for the function $h(s)$ or $h(x)$, the area calculation problem can be solved easily \bf if \rm methods of anti-differentiation can provide a $G(x)$. For some $h(x)$, this is possible. In particular, methods for anti-differentiation can be employed (sometimes) to find several functions $G(x)$ whose derivative or slope on the x-interval [a,b] coincides everywhere with the slope $h'(x)$ of $F(x)$. The problem statement above assumed one such function $G(x)$ was available. The last step in the calculation of area assumes at least one such map $G(x)$ is given or can be found.

Methods for anti-differentiation or reversing slope calculations say how to find possible formulas for a function f(x) from a single formula for its derivative (slope) m = f¢(x). These methods are ad hoc. They do not work in all examples, but they do work in a large number. Methods for finding or obtaining a function from its derivative or slope lead to formulas for the calculation of areas, volumes, weights, masses, forces, totals etc met in geometrical, physical and some business computations.

Remark. The computation of many geometric, physical and business quantities can be related to the computation of the area under the graph of some function q = h(s). The unit of area in these graphs is given by a product of the units of the horizontal and vertical coordinates q and s.

On The Definition of Functions

Functions or rules for calculating them can be introduced or defined in many ways.

1. In algebra, you may have seen the definition of functions using polynomials, square roots, small powers and so on.

2. In trigonometry, you may see the right triangle and unit circle definitions of the sine, cosine and tangent functions. Again, in trigonometry, you may see the idea of the inverse to a function employed, to define or introduce the arccos, arcsin and arctan functions.

3. The set theoretic approach to defining a function is to give a set of ordered pairs with the vertical line property. Here a finite set of ordered pairs with the vertical line property correspond to a table of values for a function.

4. In previous chapters, a function g(x) was given, introduced or defined, by the slope or derivative $f '(x)$ of another function $f(x)$
5. In this chapter, you have seen a function F(x) introduced or defined as the area-under-a curve between two points. For another example, see the definition of the natural logarithm in the next chapter.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science