Activity: Linking Integration and differentiation
Purpose
Use a numerical approach to solve problems involving summation so solve integration problems.
Define integration as the process of solving a problem by cutting it into small problems of same
type, solving the smaller problems (at least approximately) and summing the answers to get solution
of original problem. An integration problem is a problem that can be solved using this method. One
example of an integration problem is the problem of finding the area under a curve. The integral is
𝑏
the area under curve. Introduce notation for this. ∫𝑎 𝑓(𝑥0 𝑑𝑥 is area under f(x) between x=a and x=b
and emphasise the need for a shortcut to solve these problems.
Take a detour to find velocity from displacement and develop understanding of differentiation from
this. Use a numerical approach to solve problems involving finding rates of change. Find a shortcut
for finding the derivative. Develop ideas around fundamental theorem of calculus and make the link
to see integration and differentiation as inverse processes. Rate of change of an area under the
function is the height of function. Area is anti-derivative of function. (the fundamental theorem of
calculus, including: If f is continuous on [a, b] and F(x) =
using technology) and If F' = f then
for [a, b] then F' = f (explores
).
Having made connection between integration and anti-differentiation, revisit integration problems
and solve using anti-differentiation methods.
Achievement objectives
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M 8-2 Display and interpret the graphs of functions with the graphs of their inverse and/or
reciprocal functions
M 8-10 Identify discontinuities and limits of functions
M 8-11 Choose and apply a variety of differentiation, integration, and anti-differentiation
techniques to functions and relations, using both analytical and numerical methods
Indicators
Understands and uses a range of differentiation techniques including:

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first principles (only for polynomials of degree ≤ 3)
product rule, quotient rule, chain rule
other techniques including implicit or parametric differentiation (for conic relations).
Connecting derivative and instantaneous rate of change
Functions include polynomials, ex, ln(x), trig functions, xn, n ∈ R .
 Develops an understanding of integration by:

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
using numerical methods for finding areas under curves (rectangle rule, trapezium rule)
relating the definite integral to the limit of a sum (Riemann Integral)
linking integration and anti-differentiation using the fundamental theorem of calculus,
including:
Page 1
o
Part 1: If f is continuous on [a, b] and F(x) =
for [a, b] then F' = f (explores
using technology) (Note: this guarantees that an anti-derivative does exist for Part 2.)
o
Part 2: If F' = f then
 Uses a variety of integration techniques for functions including polynomials, xn ( n ∈ R), ex,
sin(x), cos(x), f ' ( g ( x ) ) ( g ' ( x ) ) ,
Applications include:



finding areas under and between curves
rates of change
kinematics.
Makes links with solving equations M7-7, M7-8, M7-9, M7-10, M8-8 and M8-12
Specific learning outcomes
Students will be able to:

Develop an understanding of integration by:
o

using numerical methods for finding areas under curves (rectangle rule, trapezium
rule)
o relating the definite integral to the limit of a sum (Riemann Integral)
o linking integration and anti-differentiation using the fundamental theorem of
calculus
Understand and use a range of differentiation techniques including:

first principles (only for polynomials of degree ≤ 3)
product rule, quotient rule, chain rule
other techniques including implicit or parametric d
ifferentiation (for conic relations).
Connecting derivative and instantaneous rate of change
Solve applications include:
o
o
o
o
o
o
o
o
finding areas under and between curves
rates of change
kinematics.
Diagnostic snapshot(s)
Students need an understanding of level 2 families of functions and graphs and algebra.
Planned learning experiences
Provide data which requires develops the need to estimate integration in context.eg finding distance
travelled given speed over time.
Initial focus is on distance, speed and time: Use the fact that distance travelled is rate multiplied by
time to find distance from speed and time.
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Progression: Find the total distance travelled from velocity and time given:
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Velocity/time data
Graph of velocity and time
Velocity as function of time (simple parabolic). Find upper and lower limits. Explore
various sub-intervals.
Velocity in terms of linear piecewise function. Check area under graph does give distance
covered.
Encourage students to use rectangles as simpler to use. To estimate upper and lower bounds,
students should divide area up into smaller rectangles that are either slightly above or slightly below
the curve.
Explore a wide variety of problems that are solved by similar method (numerical method for finding
area under curve). (Plenty of problems as traditional Simpson’s and Trapezium Rule questions.)
Define integration as the process of solving a problem by cutting it into small problems of same
type, solving the smaller problems (at least approximately) and summing the answers to get solution
of original problem. An integration problem is a problem that can be solved using this method. One
example of an integration problem is the problem of finding the area under a curve. The integral is
𝑏
the area under curve. Introduce notation for this. ∫𝑎 𝑓(𝑥0 𝑑𝑥 is area under f(x) between x=a and x=b
Lead to need for developing alternative method for doing this. Take a detour to find velocity from
displacement and develop understanding of differentiation from this. Then make the link to see
integration and differentiation as inverse processes. Rate of change of an area under the function is
the height of function. Area is anti-derivative of function. (fundamental theorem of calculus).
Revisit problems solved numerically at the start of this topic and use anti-differentiation to solve
those that you can.
Example
Task 1:
Mrs Brown and her husband are on a very long journey by car. They are travelling on a highway
that bypasses all towns, and Mr Brown had extra fuel tanks fitted to his car to extend the range to
about 1500km. Mrs Brown likes to keep track of the distance she has covered, but discovers
soon after departing from home that the tachometer (distance meter) in the car is broken, and Mr
Brown forgot to bring the road map. The speedometer still works and Mrs Brown is wearing a
watch.
1. Describe how they could determine the distance they have travelled.
2. Mrs Brown decides she will record the speedometer reading from time to time, and that she will
use these readings to estimate the distance covered from this record. The record for the first two
hours of the journey is given below. (They left home at 8:30 am). The speeds are given in
km/h.
Time
Speed
8:35
84
8:49
126
9:01
122
9:09
93
9:21
75
9:25
81
9:29
84
9:47
118
9:53
112
9:59
122
10:07
134
10:13
131
10:17
124
10:21
114
10:25
109
10:29
116
Estimate the total distance travelled.
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Task 2: Your task is to estimate the total distance travelled. Give an upper and lower bound for
your estimate. (Ie what is the greatest distance you estimate they have travelled? What is the least?
Task 3:
(c) Explore upper and lower bounds for your estimate of distance.
(d) Use differnt sub-intervals to get as accurate an estimate of distance as possible.
Task 4: Find the total distance covered when speed is described piecewise function, V. V is
velocity in kph and time is t in hours.
V = 600t;
0<t<1
V = 600;
1<t<2
V = 1200 – 300t;
2<t<4
Test the assumption that area under the graph is the same as the distance travelled.
Task 5:
Explore wide variety of problems that are solved by similar method (numerical method for finding
area under curve).
Page 4
Eg
Task 6: Find the velocity given information about distance and time.
Gowell's transport company has to make a delivery from Bloemfontein to Cape Town. The distance
between the two cities is 1041 km. The driver sets off at 5:30 and must arrive in Cape Town before
17:00, every hour that he is late the company has to pay a penalty fine. For safety reasons the
company insists that the driver take 3 half-hour stops to rest and refuel if necessary. Shortly after
leaving Bloemfontein he notices that his speedometer is not working. The tachometer still works
and the driver is wearing a watch.
1. Describe how the driver could determine when he will arrive in Cape Town.
The driver decides to record the tachometer recordings from time to time, and he will use these
readings to estimate his speed. After four hours the driver takes his first rest. The record for these
four hours is given below.
2.
2. Estimate, as well as you can, the speed of the vehicle during the first four hours.
3. Provide a written account of how you obtained your estimate.
4. If the driver managed to maintain a similar speed for the rest of the journey would he reach Cape
Town before 17:00? Explain your answer.
5. Due to the passes through the mountains in the Western Cape, the driver will have to reduce his
average speed by 10 km/h for the last 250 km. Will the driver make it to Cape Town on time?
Explain your answer.
6. During the driver’s first stop he meets you in the service station and explains his problem to you.
Could you help him to devise a plan that will give him a more accurate or easier to calculate
estimate of his speed?
Explain your plan.
Possible adaptations to the activity:
Cross curricular links
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Probability AS 3.14
Physics (kinematics, etc)
Economics (maximising profit, minimising cost)
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Surveying
Extension/enrichment ideas
For algebraically strong students, ask students to do examples of calculating areas with n sub
intervals and then sum these to generate general formula. This involves the use formulas for sums
of squares and sums of cubes.
Planned assessment
This teaching and learning activity could lead towards assessment in the following achievement
standard:


Differentiation
Integration
AS 3.7
AS 3.6
Spotlight on
Pedagogy What aspects of the pedagogy plan does this activity cover.

Planning for effective learning:
o
using learning materials to focus student attention on key concepts.

Enhancing the relevance of new learning:
o Providing appropriate levels of challenge.
o Encouraging students to explain their thinking.

Creating an inclusive learning environment
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Encouraging reflective thought and action
o
expecting all students to be able to do mathematics and statistics with understanding
 supporting students to explain and articulate their thinking
 engaging students in evaluating different methods and strategies
 getting students to justify, compare and contrast solutions
 probing student thinking

Enhancing the relevance of new learning
o exploring generalisations made by students
o encouraging students to explain their thinking
o supporting students to connect concepts and applications
o providing real-life problems in which the context is relevant to the student
Key competencies Specifics of the key competencies that would be covered by this activity – see
detailed list

Using languages symbols and texts:
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Students use the symbols and conventions of mathematics to make connections between
different representations (tables, graphs, diagrams, equations).
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Students use symbols and diagrams to solve problems.
Students interpret and communicate information and ideas, they know and use specialised
vocabulary, as well as their own language, to explain ideas.
Students interpret word problems and visual representations.
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Thinking: Students solve problems and model situations in real, mathematical and statistical
contexts. This involves:
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Students select appropriate methods and strategies when solving problems.
Students use mathematics to model real life and hypothetical situations; they make
conjectures, challenge assumptions and thinking, and they engage in sense making.
Students ask questions, want to know ‘why’, make connections and discern if answers
are reasonable.
Students seek patterns and generalisations.
Relating to others:
o Work as a group, understand others thinking, accept and value differing viewpoints,
negotiating meaning.
Values What are the links to this section of the curriculum
Students will be encouraged to value:
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
excellence, by aiming high and by persevering in the face of difficulties
innovation, inquiry, and curiosity, by thinking critically, creatively, and reflectively
Māori/Pasifika
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Trying to make links here to Maori and/or Pasifika contexts
Ka Hikitia
Compass for Pasifika success
For example: Computer-based design using traditional motifs
Planning for content and language learning what aspects of planning for content and
language learning (ESOL online) are covered in this activity, include specifics for the activity as well, see
the fourth bullet below as an example

ESOL Online

Provide multiple opportunities for authentic language use with a focus on students using
academic language:
o Is the language focus on key language?
o Do I make sure the students have many opportunities to notice and use new
language?

Reinforce vocabulary of functions, relationships and terminology of calculus.
Links
Page 7
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Introduction to differentiation
http://nrich.maths.org/4722
Connect graphs of f and f’
http://nrich.maths.org/6412
Use differentiation to carry out an an optimisation investigation exploring packaging.
Key question: What is the best package for a given shape? Start by looking at commonly
used packaging.
Connections
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Investigations in Mathematics
Investigation 3: Container design
Cooperative mathematics for level 7 Robyn MacIntyre
http://math4teaching.com

http://www.stumbleupon.com/su/9nEzMo/www.touchtrigonometry.org/
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