Arc Length and Surface Area Calculus Techniques Meet History

1
Arc Length and Surface AreaCalculus Techniques
Meet History

• David W. Stephens
• The Bryn Mawr School
• Baltimore, MD
• NCTM Baltimore 2004
• 15 October 2004

2
Contact Information

• Email
• stephensd_at_brynmawrschool.org
• The post office mailing address is
• David W. Stephens
• 109 W. Melrose Avenue
• Baltimore, MD 21210
• 410-323-8800
• The PowerPoint slides will be available on my
  school website
• http//207.239.98.140/UpperSchool/math/stephensd
  /StephensFirstPage.htm

3
Why is Arclength a Fascinating Topic?

• This is a late topic in BC Calculus.
• The seniors are getting near the end of their
  high school years and the AP exam is on the
  doorstep.
• Calculus is a great capstone course in high
  school, because it brings together all of the
  mathematics that the students have previously
  learned.

4
How is Arclength a Fascinating Topic?

• Calculus students already know about arclength on
  a circle from their geometry class
• They understand radians (although perhaps they
  still struggle with the importance of radians)
  and radians are crucial for calculus.
• It is valuable to tie in new methods to ones they
  already know. Calculus topics often lend
  themselves to doing this.

5
Calculus Strategies Integration

• The definite integral is an accumulation of
  products that is the sum of products of two
  quantities, so definite integrals can be thought
  of as measurements of areas.

6
Calculus Strategies Integration

• In any application of integration (such as areas
  under a curve, volumes, arclength, work,
  distances, or total costs), there is a three step
  strategy
• Cut the ltarea, volume, arclength, work, etcgt into
  small pieces.
• Code the quantity to be measured on a
  representative small piece, because we understand
  the geometry of the small parts.
• Recombine the parts (with sums / definite
  integrals).

7
Calculus Strategies Integration

• Step 1 (Cut the desired result into small
  pieces.)

8
Calculus Strategies Integration

• Step 2 (Code the quantity to be measured on a
  representative small piece)
• It looks like this
• dA y dx
• The width (x) is cut into infinitesimally small
  parts, and the height (y) depends on the function
  under which the area is to be measured.

9
Calculus Strategies Integration

• Step 3 (Recombine the parts with sums / definite
  integrals )
• dA y dx
• Adding up all of these simpler parts becomes
• ?

10
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 300 BC Euclidean Geometry
• Euclid (325 265 BC, probably at Alexandria,
  Egypt)
• The subject of plane geometry was known as far
  back as 2000 BC 2500 BC. Perhaps the Chinese
  and other Asian cultures knew this information
  independently at about this same time as well.
• Distance is measured with a straightedge.

History of mathematics available
at http//www-gap.dcs.st-and.ac.uk/history/BiogI
ndex.html
11
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 1629 to 1640s Cartesian coordinates
• Rene Descartes (France 1596 1650)
• Points were located with numbers, marrying
  geometry and algebra. Fermat knew these results
  in about 1629 as well.

Length is now calculated, rather than measured.
12
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 1660-1670 Integral Calculus
• Isaac Newton (1643 1727, England) and Gottfried
  Leibniz (1646 1716, Germany)
• Ideas of cutting a length into small pieces and
  measuring the small pieces with plane geometry
  methods and then recombining the pieces was a new
  strategy.
• (Details to be shown later.)

13
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 1920 1945 Measurement of Coastlines
• Lewis F. Richardson (England 1881 - 1953)
• Richardson investigated to find out that the
  reported length of coastlines in Europe (and he
  is known especially for a discussion of the
  coastline of England) varied by as much as 20.

14
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 1975 Fractals
• Benoit Mandelbrot (Poland 1924 - )
• (His family was Lithuanian Jewish. He now resides
  in the USA.)
• Methods were developed to look at the similarity
  of small pieces of a line or surface to the whole
  line or surface.
• Measurements (and the accumulation of parts of
  the measurements) seemed to depend on the scale
  of the measurement tool.

15
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 300 AD Theorems of Pappus
• Pappus ( 290-350 AD, Alexandria, Egypt)
• Pappus stated two useful theorems, long before
  the methods of calculus were in existence, which
  help to calculate volume and surface area. In
  uncanny ways, these ancient theorems are verified
  by the much newer methods of the integral
  calculus and the fractals.

16
A Whirlwind Histo-Mathematical TourHow Do We
Calculate Length?

• 1980s Gaussian Quadrature
• Texas Instruments Calculator Algorithm
• The method for performing the numerical
  integration fnInt is a fast, usually accurate,
  but complicated and fascinating algorithm.
• (This is a method used for any integration, not
  just for calculating length, but it has a
  connection to the other methods.)

17
Arclength Meets History

• Here is how a class might proceed, building up
  the ideas for calculus in a historical-mathematica
  l way.
• This discussion will proceed as if all of you are
  not actually familiar with the calculus topic of
  arclength.

18
What is an arc?
arc Middle English word derived from Latin
arcus meaning bow, as in bow-and-arrow, and,
later, arch or curve. In his 1551 Pathway to
Knowledge, Recorde used arche, arche lyne (also
spelled archline), and bowe lyne (also spelled
bowline) for the arc of a circle. Billingsley
uses the word arke in his 1570 translation of
Euclids Elements.
This is from Historical Modules for the teaching
and Learning of Secondary Mathematics (December
2002, Mathematical Association of America). This
definition comes from Lengths, Areas and
Volumes (page 193).
19
What do little pieces of most functions look like?

• Most functions have a curve to them, so the
  question of the length of an arc amounts to
  calculating the length of a piece of a function.
• Calculus students have been well trained to say
  that little pieces of most functions look like .
• line segments, because functions are usually
  locally linear.

20
Setting up Arclength

• So calculating the length of a curve comes down
  to methods to measure the length of a line
  segment.
• Cut the ltarea, volume, arclength, work, etcgt into
  small pieces.
• Code the quantity to be measured on a
  representative small piece, because we understand
  the geometry of the small parts.
• Recombine the parts (with sums / definite
  integrals).

21
Setting up Arclength Now we follow the history

• Pythagoras (569 475 BC, Samos, Ionia)
• No coordinates available

22
Setting up ArclengthUse Pythagoras in a
calculus class

• We want to know the length of y x2 on the
  interval 0 , 4.
• We do this in four pieces to begin.

23
Setting up Arclength Use Pythagoras in a
calculus class
Here are the four triangles whose hypotenuses are
straight, under the assumption that curves are
locally linear.
So s about 16.747
24
Setting up Arclength Use Pythagoras in a
calculus class

• Notice that we have
• (1) cut the curve into small pieces even though
  Pythagoras would not have understood the idea of
  a function with coordinates,
• (2) used the geometry of Pythagoras to calculate
  the lengths of the four pieces, and
• (3) recombined with addition. No calculus was
  used, but the ideas of calculus were employed.

25
Setting up ArclengthAdd Descartes to the
question

• For each of the triangles, coordinates are used
  to locate the points on the function, and the
  distance formula is that of Pythagoras with
  adaptations for the coordinates.

26
Setting up Arclength Add Descartes to the
question

• The coordinates of the points are
• (4, 16) ,
• (3 , 9) ,
• (2 , 4),
• ( 1, 1),
• and (0, 0)

27
Setting up Arclength Add Descartes to the
question

• Using the distance formula on each of the
  triangles gives the same results as before.

28
Setting up Arclength A Detour to the 20th
Century

• Calculus students accept the idea of local
  linearity fairly easily, even though it is a
  novel idea at first.
• To challenge their acceptance of this idea (and
  recall it is late in the senior year at the end
  of a long and challenging AP course), lets move
  to Richardson and Mandelbrot and the coastline
  of England (and other places).

29
Length of a Coastline
From Chaos by James Gleick (Penguin Books 1987,
page 95)
30
Length of a Coastline

• Some of the stories told about the measurement of
  coastlines include the importance of knowing the
  length of the coastlines of England and Norway
  during World War II, so that the navies knew how
  long a coastline they needed to defend.
• Later it became a fascinating mathematical topic.

31
Length of a Coastline

• We can actually do some measurements now to see
  how this paradox of Lewis Richardson goes.
• We will simulate this with the maps of Jaggedland
  and Smootherland
• We can measure with different smallest units
  available.

32
Length of a Coastline

• Use a 3 inch straightedge.
• Start at some point on the map.
• Swing the 3 inch straightedge until it first hits
  another point on the map.
• Move the end of the 3 inch straightedge until it
  is at the last endpoint
• Count how many 3 inch measurements you can make,
  continuing until you are back at the starting
  point.

33
Length of a Coastline

• Use a 1 inch straightedge.
• Do the same process as above.
• Use a ½ inch straightedge.
• Do the same process as above.
• Use the scale on the map to convert the total
  number of inches to miles.

34
Length of a Coastline
Use actual maps of Florida, Norway, England, the
Chesapeake Bay, and the Mississippi River in
classes.
A Student Worksheet
Observations
35
Length of a Coastline

• Actual mileages whatever actual means (since
  we are now skeptical about whether there is a
  real answer ????)
• Florida .1,350 miles
• England 5,581miles (6261 including islands)
• (11,072 miles for Great Britain, 19491 including
  islands)
• Norway
• Chesapeake Bay 11,864 miles of shoreline
• Mississippi River 2,350 to 2,552 miles
• (depending on who you ask)

36
Length of a Coastline

• What seems to be the results and connections?
• As the measuring tool gets shorter, the total
  length gets longer, but not always!
• What measurement tool does a geological survey
  use? Why?
• Actual length seems to be the result of practical
  methods, but they are not definite answers.

37
Length of a Coastline

• Small pieces on the maps are measured as the
  Greeks would have done it (!!), and the
  Pythagorean theorem could have been used to
  calculate from the vertical and horizontal.
• Old meets new.
• Mathematics is still evolving and new methods and
  ideas are still being added.
• It is OKAY to combine new and old ideas!

38
Length of a Coastline

• Coastline Paradox
• Determining the length of a country's coastline
  is not as simple as it first appears, as first
  considered by L. F. Richardson (1881-1953). In
  fact, the answer depends on the length of the
  ruler you use for the measurements. A shorter
  ruler measures more of the sinuosity of bays and
  inlets than a larger one, so the estimated length
  continues to increase as the ruler length
  decreases.
• In fact, a coastline is an example of a fractal,
  and plotting the length of the ruler versus the
  measured length of the coastline on a log-log
  plot gives a straight line, the slope of which is
  the fractal dimension of the coastline (and will
  be a number between 1 and 2).
• from http//mathworld.wolfram.htm

39
Length of a Coastline

• How Long is the Coast of Great Britain?
•   Figure 1 The coastline of Great Britain In
  1967, Benoit Mandelbrot published 7 How Long
  is the Coastline of Great Britain'' in Nature. In
  it, he posed the simple question of how one
  measures the length of a coastline. As with any
  curve, the obvious answer for the mathematician
  is to approximate the curve with a polygonal
  path, each side of which is of length ? . (See
  Figure 2.)
• Then by evaluating the length of these polygonal
  paths as ??0 , we expect to see the length
  estimate approach a limit. Unfortunately, it
  appears that for coastlines, as ??0 , the
  approximated length L(?)? infinity as well.

  Figure 2 Approximating the coastline of
Great Britain
40
Length of a Coastline

• In a later book, 10, pp 28-33, Mandelbrot
  discusses the extensive experimental work on this
  problem which was done by  Lewis Fry Richardson.
  Richardson discovered that for any given
  coastline, there were constants F and D such that
  to approximate the coastline with a polygonal
  path, one requires roughly Fe-D intervals of
  length . Thus, the length estimate can be
  given as L(?) Fe 1-D
• The reason has to do with the inherent
  roughness'' of a coastline. In general, a
  coastline is not the type of curve we are usually
  used to seeing in mathematics. Although it is a
  continuous curve, it is not smooth at any point.
  In fact, at any resolution, more inlets and
  peninsulas are visible that were not visible
  before. (See Figure 3.) Thus as we look at finer
  and finer resolutions, we reveal more and more
  lengths to be approximated, and our total
  estimate of length appears to increase without
  bound.

http//www.math.vt.edu/people/hoggard/FracGeomRepo
rt/node2.html
41
Length of a Coastline

• Contrast this idea with the foundations of
  calculus which assert that a limit is attained
  when we cut the length into smaller and smaller
  pieces.
• We make the assumption and conclusion that
  there is a finite length and that our methods of
  the integral calculus will help calculate that
  length.

42
Length of a Coastline

• What is the length of the coastline of Britain?
  Benoit Mandelbrot proposed this question to
  demonstrate the complexity of measurement and
  scale. There are a number of almanacs that
  provide this information. However, if one
  examines the measuring techniques used to
  determine the length of Britain's coastline, it
  becomes obvious that this measurement is only an
  estimate based on the accuracy of the measuring
  device. Smaller units mean greater accuracy. But
  we can continue that line of thinking
  indefinitely, just as we do with fractions. There
  are always smaller fractions, an infinite number.
  Therefore, the coastline of Britain is an
  infinite length, however, it is confined within a
  finite space. We can begin to understand then
  that perimeter can have an infinite length
  confined within a finite area.

http//home.inreach.com/kfarrell/measure.html
43
Length of a Coastline

• How Long Is Australia's Coastline? (an
  explanation)
• At first blush the question seems eminently
  reasonable, but it is as open-ended as the
  classical "how long is a piece of string?" The
  answer to both is the same it all depends.
• Dr Robert Galloway of the CSIRO Division of Land
  Use Research in Canberra was recently confronted
  with the question when compiling an inventory of
  Australia's coastal lands. Looking up the
  published figures he found the following answers
  
• The great disparity has to do with the precision
  with which the measurement is made. The larger
  and more detailed the map, and the more finely
  the measurement is made, the longer will be the
  coastline. Ultimately one could walk around the
  coast itself with a measuring stick, but the
  answer still depends on whether you use
  seven-league boots or a metre rule.
• (It's a philosophical point whether the coastline
  tends to any limit as precision improves. Some
  say it does, others not.)
• To settle on a reliable, repeatable figure, Dr.
  Galloway got together 162 maps covering the
  Australian coast and enlisted the help of Ms
  Margo Bahr of the Division.

44
Length of a Coastline

• A few points of methodology had to be agreed on
  before the exercise could begin
• How far up estuaries should the coastline be
  taken? It was decided that all inlets would be
  arbitrarily (but consistently) cut off whenever
  their mapped width was less than 1 km. Within
  Sydney Harbour, for example, Kirribilli Point was
  joined to Garden Island. Straits less than 1 km
  wide were ignored, treating the island as though
  it were part of the mainland.
• Islands less than 12 ha. were ignored. Measuring
  the coastline of the 2600 islands larger than
  that would be tedious in the extreme. Instead, a
  16 sample was taken and a graph of coast length
  against area drawn.
• This plot gave a good correlation, allowing
  island coastlines to be derived simply from their
  area. However, the ten largest islands (including
  Tasmania) were, for accuracy, measured directly.
  (Macquarie Island and Lord Howe Island were not
  included.)
• Mangroves were regarded as part of the land, with
  the coastline following their seaward fringe
  channels between mangroves were treated as
  estuaries. All coral reefs were excluded.

http//www.maths.mq.edu.au/numeracy/tutorial/cts2.
htm
45
Length of a Coastline

• Finally came the question, which tools to use a
  pair of dividers, a map measuring wheel, or a
  length of string or fine wire? On a test run,
  dividers gave consistent results only if the same
  starting point was used the wheel was rapid but
  inaccurate. Fine wire (not string) laid on the
  drawn coastline proved surprisingly consistent
  and accurate (as good as dividers set to a 0.7 km
  interval) and so was chosen for the task. Down to
  work!
• When the 162nd map was put aside, the total
  length of the mainland coast plus Tasmania worked
  out to be 30 270 km. Adding on the length of the
  coast of all the islands greater than 12 ha,
  about 16 800 km, gave a grand total of 47 070 km.
  
• As a matter of interest and undeterred by their
  prior efforts, the two workers examined the
  effect of different divider lengths on the
  measured coastline. As expected, the apparent
  length of the coast of the mainland diminished
  steadily as the divider length was increased
  shrinking to 10 830 km at a 1000-km intercept.
• A simple formula was derived that linked coast
  length to the measuring intercept. Using this
  formula to extrapolate a divider length of just 1
  mm, gave a length of about 132 000 km for the
  mainland of Australian rather more than three
  times the circumference of the earth!

46
Length of a Coastline

• How Long Is Australia's Coastline?
• The correct answer is .... it depends!
• It depends on which source you read, apparently.
• Source Length
• Year Book of Australia (1978) 36 735 km
• Australian Encyclopedia 19 658 km
• Australian Handbook 19 320 km
• So who is correct? The answer is all of them!
  Each source used a ruler with different sized
  increments on it. If you measure the coastline
  with a ruler that is just 1 mm long, you would
  get a length of 132 000 km!

47
Length of a Coastline

• 4.2 Calculating coastline and population in New
  Zealand for Maori tribes
•  
• 4.2.1 Coastline calculations
• Under the proposed allocation method, inshore
  fishstocks and 60 of deepwater fishstocks would
  be allocated according to the length of an Iwis
  coastline. Exactly how would coastline lengths be
  worked out?
• It is proposed that a 150,000 scale map of New
  Zealand would be used. Iwi would have to reach
  agreement with neighbouring Iwi as to their
  respective coastline lengths. The exact coastline
  length for a quota management area would then be
  calculated as follows
• rivers would be cut off at the coast and the
  distance across the river mouth included in the
  coastline measurement
• the coastline length of harbours and bays whose
  natural entrance points are greater than 10 km
  apart would be included in the coastline
  measurement
• the juridical bay formula (see below) would be
  applied to harbours and bays whose natural
  entrance points are less than 10 km apart in
  order to determine whether those harbours and
  bays would be included in the coastline
  measurement and
• with the exception of the islands in the Chatham
  Islands group, coastline measurements would not
  include the coastline of islands claimed by Iwi
  to be part of their traditional takiwa.

48
Length of a Coastline

• The juridical bay formula
• The juridical bay formula is applied to bays
  where the natural entrance points are less than
  10 km apart in order to determine whether the
  distance across the entrance of the bay or the
  actual coastline of the bay should be added to
  the coastline measurement. The formula works as
  follows (see Figure 3)
• a straight line is drawn between the natural
  entrance points of the bay
• a semicircle is drawn on the straight line (using
  the straight line as the diameter of the circle)
  and the surface area of the semicircle is
  calculated
• the surface area of the bay enclosed by the
  straight line is also calculated using map
  information software
• if the surface area of the semicircle is smaller
  than the water surface area of the bay, then the
  distance between the natural entrance points is
  included in the coastline measurement, (see
  figure 3a, the shoreline of the bay is not
  included) or
• if the surface area of the semicircle is bigger
  than the water surface area of the bay, then the
  shoreline of the bay is measured and included in
  the coastline measurement.( see figure 3b)

49
Length of a Coastline

• A drawing of the New Zealand juridical bay
  formula

http//www.tokm.co.nz/allocation/1997/implementing
.htm
50
Length of a Coastline

• The Florida Shoreline and its Measurement
• The FLDEP is responsible for monitoring and
  managing approximately 680 miles of Floridas
  coastline.
• This includes the states entire coastline except
  for Monroe County (Florida Keys) and Federal
  sites. These management efforts are the result of
  two contributing factors. In addition, nature
  constantly changes the shoreline through normal
  coastal processes and occasional storm events.
  Other shoreline changes result from mans
  engineering activities associated with ports and
  harbors and shoreline stabilization. The FLDEP
  must identify and quantify these changes and
  manage the coastline to preserve Floridas most
  important natural resource its beaches.

51
Length of a Coastline

• Traditional Method to Measure the Florida
  Shoreline
• Measurements are taken at fifty feet intervals
  and at points of slope change along these cross
  sections. The nearshore survey extends from the
  waterline to either the 30 ft contour line or
  2,400 ft from the shoreline, whichever is closer.
  Survey monuments on the baseline are maintained
  so that subsequent surveys may be taken in the
  same locations providing a common reference point
  to allow comparisons. Because of the time
  required to collect these data, only three to
  four counties shorelines are surveyed each year
  using this technique. Aerial photography is taken
  in conjunction with the surveys to provide visual
  record and supplement the coastline management
  process.

http//64.233.161.104/search?qcache77CGD4GeIfgJ
www.thsoa.org/pdf/hy99/9_3.pdfcoastlinemeasureme
ntFloridahlen
52
Length of a Coastline

• Advancements in survey technology have provided
  new tools for gathering data to manage the
  coastline. Airborne lidar is one such
  advancements. Lidar is an acronym for LIght
  Detection And Ranging. Lidar works similar to
  radar, but a laser is used instead of radio waves
  for distance measurements. Each laser pulse is
  transmitted from the airborne platform to the
  surface below. Some of the light energy is
  reflected from the water surface and detected by
  onboard optical sensors. The remaining energy
  continues through the water column, is reflected
  from the bottom and is detected by the onboard
  sensors. The time difference between the two
  energy returns indicates the water depth.

53
Length of a Coastline

• The traditional methods would require about 2
  years time to complete a measurement of the
  Florida coastline, where the new methods require
  about 1 month.
• though the use of airborne lidar the areas were
  surveyed in approximately one month producing
  data on an 8m by 8m grid spacing as opposed to
  the historic 1,000 ft cross sections. The speed
  and cost effectiveness of SHOALS will enhance
  FLDEPs ability to resurvey more beach areas on
  a more frequent basis

54
Length of a Coastline

• The sophistication of methods, as well as the
  extraordinary effort that is expended to measure
  coastlines, indicates to students that
• (1) the process is important,
• (2) the assumptions about how to accomplish the
  measurement are heeded, and
• (3) mathematical methods and practical
  considerations are both part of the process.

55
Arclength using calculus

• Now we get to the calculus methods to measure
  arclength.
• Step 1 Cut the length of the curve into small
  pieces. (Small is undefined)
• We get little triangles, and we still believe
  that the curves are usually locally linear, so
  that the hypotenuse is very close to linear.

56
Arclength using calculus

• Step 2 Code the quantity to be measured on a
  representative small piece, because we understand
  the geometry of the small parts

57
Arclength using calculus

• Step 3 Recombine the parts (with sums / definite
  integrals)

58
Arclength using calculus

• This is not quite in the form that we prefer,
  since we need a dx or dy outside the square root.
  So

59
Arclength using calculus

• Similar derivations from the original
• can be done by dividing by either (dy)2 or (dt)2
  to get the companion arclength formulas

60
Examples of arclength

• Lets look at an example
• f(x) x2 on 0 , 4
• S

61
Examples of arclength

• Heres another example
• For f(x) sin -1 (x), suppose that students have
  not learned how to integrate the arctrig
  functions.
• So y sin -1(x) becomes x sin y
• S

62
Examples of arclength

• Heres an example using parametric functions
• x cos(t)
• y sin(3t)
• S

63
Surface area Back to the past

• There is an interesting way to remember how to
  calculate surfaces area in calculus that relies
  on a very old geometry idea.
• We look at the ideas of Pappus (300 AD, Egypt)

64
Theorems of PappusArea and Volume Arclength
and Surface Area

• Theorem 1
• If a region is rotated about an axis that does
  not intersect with the region, then the volume
  generated equals the product of the area of the
  cross section of the region and the distance that
  the center of that cross section travels.
• V
• So dV or
• These are the shell methods of volumes of
  revolution.

65
Theorems of PappusArea and Volume Arclength
and Surface Area
Rotate the yellow region about x -1. Pappus
says that the volume generated by the black strip
is dV
66
Theorems of PappusArea and Volume Arclength
and Surface Area

• Theorem 2
• If a arc is rotated about an axis that does not
  intersect with the arc, then the surface area
  generated equals the product of the length of the
  arc and the distance that the center of that arc
  travels.
• SA
• The radius is determined by the axis about which
  the arc is rotated (and there are the other two
  arclength formulas mentioned earlier)

67
Theorems of PappusArea and Volume Arclength
and Surface Area

• It seems that this second theorem works perfectly
  well as a geometry formula

Since we get the lateral area of a cylindrical
prism.
68
Theorems of PappusArea and Volume Arclength
and Surface Area

• but it seems a bit suspicious for an arc that is
  not perpendicular to the axis of rotation

But remember that arc AB is very small.
69
Surface Area (examples)
Calculate the surface area when the curve f(x)
x2 on 0 , 4 is rotated about the y axis. SA

whose value can be done with a u du substitution
or a fnInt on a calculator.
70
Surface Area (examples)
Calculate the surface area when the curve f(x)
x2 on 0 , 4 is rotated about the x axis. SA
71
Surface Area (examples)
Calculate the surface area when the curve f(y)
sin(y) on 0 , is rotated about the x
axis. SA
The numerical integral is needed to calculate
this value. fnInt(
, y, 0 , )
72
Surface Area (examples)
Calculate the surface area when the curve defined
by x cos(t) and y sin(t) is rotated about the
x-axis. Use 0 t
SA

73
Surface Area (examples)
Note that we have just calculated the surface
area of a sphere. V SA
74
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• Now, we are in the mid to late 1980s with
  technology.
• The Texas Instruments family of calculators has a
  fancy method to evaluate the definite integrals
  which is fast and usually quite accurate.
• Their website describes it as a version of
  Gaussian quadrature.

75
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• TI-S2 FAQ Tl-83 f AO TI-8S FAQ TI-86
  FAQNumeric integration on the TI-82, TI-83,
  TI-85, TI-86 - how does it work?
• The method used in the TI-82, 83, 85, and 86 is
  known as Gauss-Kronrod integration. It is a good
  deal more sophisticated than commonly known
  methods like Simpson's rule or even Romberg
  integration. In fact, the widely used mainframe
  library program QUADPACK uses this technique. A
  detailed discussion is much better left to the
  literature on this subject, which is voluminous.
  A few comments, however, may help you in
  utilizing this function.
• The concept begins with Gaussian quadrature
  rules, which have the property that by sampling a
  function at "n" points, an integral can be
  estimated that would be exact if the function
  were a polynomial of degree "2n-l". For
  comparison, the trapezoid rule is of polynomial
  degree one and Simpson's rule is of polynomial
  degree 3. But while Simpson's rule gives an exact
  integral for cubic polynomials based on three
  points, a Gauss quadrature rule with three points
  will give the exact integral of polynomials up to
  order 5. The next concept is to use a pair of
  Gauss quadrature rules of different order to
  provide both an integral result and a
  corresponding error estimate. The development of
  optimal extension pairs (pairs for which the
  lower order sample points are a subset of the
  higher order ones so that extra function
  evaluations are avoided) was done by A.S. Kronrod
  in 1965. The use of these pairs allow the method
  to be "adaptive". The integral is developed by
  beginning with the whole interval, computing the
  integral and the error estimate. If the error is
  too large, the interval is bisected and the
  process repeated on each half. Anytime a segment
  passes with regard to its "share" of the total
  error budget, that integral is added to the total
  integral. Other segments that don't pass are
  further divided. Note that this is quite
  different than some "adaptive" methods that
  iterate with smaller intervals until the result
  changes by less than some tolerance. Our method
  computes error from two integral estimates and
  only takes new function samples in rapidly
  changing regions that require it.

Formerly on http//www.ti.com/
76
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• From a user standpoint, this means that you need
  to specify an error "tolerance" that you want to
  achieve for the integral. In our implementation
  this is an "absolute" error bound. This means the
  algorithm will quit when the absolute value of
  the difference in its two integrals is less than
  the tolerance you specify. Hence, you may not
  want to ask for a tolerance of .00001 if you
  compute an integral that has a value of order
  10,000 because you are asking for 10 accurate
  digits. Another thing to watch out for is the
  fact the function is sampled a finite number of
  times on the first integration attempt. These
  points are not equally spaced and are in fact
  clustered toward the end points. However, it is
  not uncommon to have a function that is virtually
  constant or perhaps zero throughout much of the
  range for integration. In this case, the
  integrator may quit early and fail to detect the
  existence of function behavior in other regions.
  An example of this is
• fn!nt(eA(-t/le-6),t,0,l). This may seem
  appropriate to an electrical engineer who wants
  to compute all the "energy" in this waveform with
  a microsecond order fall time. However, just over
  a value of 0.002 in the interval 0 to 1, the
  value of this expression becomes less than le-999
  and underflows to zero. So the algorithm returns
  a value of zero for this integral. A range of (0
  , 0001) would be more appropriate for the problem
  and also because the integral is small, a tol
  tolerance of le-10 is not excessive.
• References
• "Numerical Methods and Software", David Kahaner,
  Cleve Moler and Stephen Nash, Prentice Hall,
  1989, Chapter 5.

77
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• It works something like this
• The interval along the x-axis is divided into two
  equal pieces.
• The area under each half is calculated.
• The half-intervals are each cut in half again,
  and the area is calculated again.
• If the area originally calculated does not change
  when the interval is subdivided, then it is
  assumed that the area is close to correct.

78
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• If the area is significantly different (whatever
  significantly different is), then it is further
  subdivided again and again until one calculation
  is not significantly different than the preceding
  one.
• Thus, parts of a function are divided more than
  other parts. (Which parts?)

79
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• However, in exchange for the accuracy (and often
  the speed) of the calculations, there are
  occasional examples for which the calculators
  make BIG errors, or for which answers are not
  available.
• Ex fnInt(1/x , x , 1 , 1 x 1020)
• Ex fnInt(sin(x) , x , 0 , 1000) is really
  slow.
• Ex fnInt(sin(x) , x , 0 , )
• gives 2.718 x 10-9 for an answer, which is only
  slightly different from the exact answer of 0.

80
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals

• Ex fnInt(e(-10x), x, 0, 10000)
• gives 0, when the actual answer is 0.1
• There is a similar strategy similar to the
  coastline measurement being used, but the entire
  function is not sampled if changes are not
  detected earlier enough, so major errors in
  calculation can occur.

81
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals
1 x lt 2 f(x) 1001
2 lt x lt 3 1 x gt 3
Calculators can accept these piecewise defined
functions
82
Gaussian Quadrature How the TI calculators
evaluate numerical definite integrals
The area under the curve on 0, 1000 ought to be
2000, but fnInt (Y1 , x , 0 , 1000) gives 1000.
1 x lt 2 f(x) 1001
-2 lt x lt 3 1 x gt 3
83
In conclusion

• Arclength and surface area may not be the most
  crucial topics in an AP Calculus class.
• but there are some connections with ideas and
  the historical development that make the topic
  fascinating and valuable.

84
In conclusion

• We followed the ideas of measuring a straight
  line from the methods of Pythagoras, which the
  students learned in algebra and geometry classes.
  So the new calculus methods have pedagogical
  ties to previous knowledge.
• We connected the geometry of the Pythagorean
  theorem to the algebra of the coordinate plane.

85
In conclusion

• We help students to make the leap to the
  measurement of an arc as a series of line
  segments, because of the local linearity of most
  functions.
• This develops the official methods of measuring
  arclength.

86
In conclusion

• But.the anecdotes from history bring some
  authentic relevance to why these lengths are
  valuable
• Geography in the 20th century use calculus ideas
  to measure coastlines.
• Calculator technology adapt issues that plague
  coastline measurement, developing highly
  effective ways to evaluate definite integral.
• Calculus improves the methods of algebra and
  geometry to do effective calculations.

87
In conclusion

• Students often cannot help but be eager to learn,
  especially when the intellectual twists and turns
  are authentic and unexpected.