Chapter 17. Area Approximation
Volume 3, Why Slopes and More Math.
Saying precisely how to compute a quantity defines it.
Covering A Region by Squares
The areas of squares and rectangles may be calculated by calculating the
product of the lengths of their sides. In the plane, the area of a
bounded region $S$ rectangular or not, may be approximated by covering
the region $S$ concerned with small squares, all of the same size,
overlapping, if at all, only at their edges.
The example of an elliptic shaped region $S$ is shown. Each covering by
small squares gives three methods for approximating what the area $A$ of
the region should be.
-
An inner and lower/under approximation to the area $A$ of the region $S$
can be obtained by summing up the areas of all the squares contained
completely in the region $S$. This inner approximation is expected to
yield an estimate \em lower \rm or $\le A$.
-
An outer and upper/over approximation of the area can be obtained by
summing up the areas of the squares which have an interior point in
common with the region. This outer approximation is expected to yield
an estimate higher or $\ge A$. A point in a square but not on an
edge is said to be an interior point of the square.
-
A middle approximation might be obtained by adding to the inner
approximation, the areas of those square which are completely in or
more than half-in the region $S$. Other inbetween approximations are
possible. Intermediate approximations yield an middle area estimate
between the upper and lower estimates.
From a computational perspective, more than half-in but not
completely in is not easy to define. This could be a matter of visual
judgment -- a step outside of the domain of rule-based mathematics.
To give a mathematical algorithm, the toss of a coin might be
sufficient, or a judgment could be made on how many of the four
triangles formed by the diagonals are included completely in the
region $S$.
One or more of the above approximations may be familiar to you from your
elementary school days.
Each of the above approximations is expected to improve as the squares
are quartered (their sides halved) repeatedly and indefinitely. The
latter would cause the lower estimate to increase, the upper estimate to
decrease while the middle estimate together with the area $A$ presumably
approximate, remaining inbetween. Such halving results three sequences of
numbers or quantities.
The lower estimates yield the increasing sequence, the upper estimate
yield the decreasing sequence, and the middle estimates yield a
sequence between the previous two.
The area $A$ should be the common, finite, limiting value $L$ of the
approximations as the sides of the covering squares become smaller
(approach zero). This says how to compute the area $A$ with an unlimited
accuracy if a common, finite limiting value $L$ exists for the
approximations.
The area of a region is defined by the methods for approximating it. That
is, the region has an area $A=L$ if and only if the three numerical
approximations described above all approach a single finite limiting
value $L$. This limiting $L$ is then called the area of the region.
Otherwise, with some disappointment perhaps, we may say that the area is
not defined. (Alternatively, we might define inner and outer areas using
the limiting values of the inner and outer approximations and identify
circumstances in which they are equal.
Optional (Advanced Material:
For some odd (pathological) region, inner and outer area approximations
may approach an inner limit $L_{inner}$ and an outer limit $L_{outer}$
which are not equal. Regardless, these limiting values may define what
is meant by inner or outer concepts of area. There has been a concern
with identifying conditions in which inner and outer approximations etc
approach the same limits. For more information, a very specialized
(that is, not for everyone) book or course on area computation, more
precisely, integration theory, is indicated.
|
|
Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|