Analytical ToolboxAnalytical Toolbox

Integral Calculus

ByBy

Dr J.P.M. WhittyDr J.P.M. Whitty

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Learning objectivesLearning objectives

• After the session After the session youyou will be will be able able to:to:• Define integration in terms of the Define integration in terms of the

anti-derivative anti-derivative • Integrate Integrate simplesimple algebraic algebraic

functionsfunctions• Use the fundamental theorem of Use the fundamental theorem of

integral calculusintegral calculus• Use math software to solve Use math software to solve

simple integration problemssimple integration problems

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Integral CalculusIntegral Calculus "It is interesting that, contrary to the customary "It is interesting that, contrary to the customary

order of presentation found in our college order of presentation found in our college courses where we start with differentiation courses where we start with differentiation and later consider integration, the ideas of and later consider integration, the ideas of the integral calculus developed historically the integral calculus developed historically before those of differential calculus. before those of differential calculus.

Some time later, differentiation was created in Some time later, differentiation was created in connection with problems on tangents to connection with problems on tangents to curves and with questions about maxima and curves and with questions about maxima and minima, and still later it was observed that minima, and still later it was observed that integration and differentiation are related to integration and differentiation are related to each other as inverse operations"each other as inverse operations"

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IntegrationIntegration

Today we view integration as the Today we view integration as the inverse of differentiation and we work inverse of differentiation and we work from the premise of gradients of from the premise of gradients of tangents to derive the equations of tangents to derive the equations of curves.curves.

We also try to establish an algorithm We also try to establish an algorithm that will make it easy to work that will make it easy to work backwards. To differentiate a polynomial backwards. To differentiate a polynomial function function ff((xx), we use the notation), we use the notation

1)('then

)( If

n

n

nxxf

xxfy

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Inverse of Inverse of differentiationdifferentiation

We can relate a given derivative with the We can relate a given derivative with the corresponding corresponding f(x)f(x). This asks you to . This asks you to work in reverse order from differentiating work in reverse order from differentiating problems, where you were given problems, where you were given f(x)f(x) and asked to derive and asked to derive f '(x)f '(x). .

This time, you have been given the This time, you have been given the derivative and asked to work back up the derivative and asked to work back up the chain - what function produced this chain - what function produced this derivative?derivative?

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Lemma:Lemma:

Instead of subtracting form the Instead of subtracting form the index we must add and instead of index we must add and instead of multiplying by the index we must multiplying by the index we must divide by the new index. divide by the new index.

This leads us to the Lemma.This leads us to the Lemma.

1)(then

)(' If

1

n

xxf

xdx

dyxf

n

n

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Integration of standard Integration of standard functionsfunctions

It is usual in calculus It is usual in calculus textbooks that to see textbooks that to see tables of standard tables of standard functions and their functions and their respective integrals respective integrals

• For this For this introductory introductory course you will course you will only require the only require the followingfollowing

y ydxdxxf )(

c cx

nax 1/1 naxn

axsin axa cos1

axcos axa sin1

axeax expa

ax

a

eax exp

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Class Examples TimeClass Examples Time

Copy and complete Copy and complete the following table.the following table.

YOU MUST ALWAYS YOU MUST ALWAYS REMEMBER THE REMEMBER THE CONSTANT OF CONSTANT OF INTEGRATION!!INTEGRATION!!

y ydx

c0

cx 33

cx 24x8

cx 45320x

cxx 23 329 2 x

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Class examplesClass examples

1.1. Integrate the following Integrate the following

2.2. Find the following integralsFind the following integrals

11812 23 xxy

7494 23 xxxy

xdxexx x sin103cos98 23

xexxxxxf

dxxf

42 284sin123sin66)4exp(36)( where

)(

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Class examples solutionsClass examples solutions

1) Write as1) Write as

cxxxcxxx

dxxxydx

33133

23

631

1

3

18

4

12

11812

cxxxx

dxxxxdxy

723

7494

234

23

and

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Class examples solutions Class examples solutions cont…cont…

2) Just find the anti-derivatives thus: 2) Just find the anti-derivatives thus:

cxdxexx

xdxexx

x

x

cos3sin32

sin103cos98

22

102

23

cexxxe

dxexxxx

xx

x

434

42

74cos32cos229

284sin123sin66)4exp(36

and

Each time we MUST REMEMBER the +c

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Further ExamplesFurther Examples

Integrate the following:Integrate the following:

a)

b)

c)

d)

e)

xxxdxx sin33cos2cos33sin6

xexxdxxxx 771ln25cos2)7exp(/25sin10

32 34sin23cos394cos83sin15 xxxdxxxx

xxxdx 5sin5/5sin55cos5

xx exdxex 22 2)ln(4/1

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Theorem:Theorem:

The fundamental theorem of calculus:The fundamental theorem of calculus:

Basically states that the area under Basically states that the area under anyany curve can be found via integration curve can be found via integration and application of limits, thus:and application of limits, thus:

ba

y

F(a)F(b)

F(a)F(b)ydxxFydxb

a

and between

function under the area the

is of value theand

then )( if

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Area of a triangleArea of a triangle

We will illustrate this theorem via a We will illustrate this theorem via a elementary example. Consider a 45 elementary example. Consider a 45 degree right angled triangle as shown.degree right angled triangle as shown.

x

y

1

We know that the area of this triangle is ½ square units. However we wish to prove this though integration to demonstrate the theorem

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Fundamental theorem of Fundamental theorem of integral calculus:integral calculus:

First we write First we write yy as a function of as a function of x. x. IN IN THISTHIS case we have: case we have: y=x.y=x. Then we Then we set up the integral with the correct set up the integral with the correct limits i.e. 0 and 1 in this case.limits i.e. 0 and 1 in this case.

1

0

xdxydxb

a

2

1

2

0

2

1

2 :gives limits theEvaluating

221

0

2

x

1

0

21

0 2

xxdx

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ExampleExample

Find the area under the curve the x Find the area under the curve the x axis and when x=1 and x=3axis and when x=1 and x=3

215 2 xy

3

1

231 215 dxxArea 3133

1 35 xxArea

13153335 3331 Area

12431 Area Square

units

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Area between curvesArea between curves

The theorem is useful when deteriming The theorem is useful when deteriming the areas between two curves. Here we the areas between two curves. Here we simply subtract the larger integral from simply subtract the larger integral from the smaller one. For example.the smaller one. For example.

Find the area between the two curves:Find the area between the two curves:

2 and xyxy

The only trick here is to find where the curves cross in order to find the limits of integration

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Solution:Solution:

Solve the problem simultaneously to Solve the problem simultaneously to find the limits of integration:find the limits of integration:

1or 0)1( xxx

Now find the integrals:

21

1

0

21

0 2

xxdx

6

1

3

1

2

1

022 xxxx

31

1

0

31

0

2

3

xdxx

The result is of course the big area minus the little one:

Square Units

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Use of mathematic Use of mathematic SoftwareSoftware

As with differentiation these days we As with differentiation these days we are able to solve problems involving are able to solve problems involving integral calculus and the integral calculus and the fundamental theorem by utilizing fundamental theorem by utilizing the MATLAB symbolic toolbox. So the MATLAB symbolic toolbox. So you can you can always always check your answers check your answers prior to handed in assignment work prior to handed in assignment work using this method. Consider the using this method. Consider the previous example.previous example.

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MATLAB: MATLAB: Integration Integration

The process is the same as usual. i.e. easy as ABC!

)int(y2,0,1-)int(y1,0,1

x^2y2 x;1y

xy2 y1 syms

:Commands MATLAB

A. Set up your symbolics in MATLAB using the syms command

B. Type in the expression remembering the rules of BIDMAS

C. Use the appropriate MATLAB function in this case int( ), making pretty if required.

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MATLAB SolutionMATLAB Solution

This is really easy This is really easy you can even you can even ask MATLAB to ask MATLAB to solve for the solve for the limits for you limits for you using the solve using the solve command if you command if you wish. This is left wish. This is left to an exercise to an exercise from previous from previous work!work!

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More MATLABMore MATLAB

Note it is also Note it is also possible to possible to evaluate evaluate indefinite indefinite integrals but you integrals but you have to put the have to put the constant in constant in yourself. Here yourself. Here you simply don’t you simply don’t put in the put in the integration limits, integration limits, e.g.:e.g.: xdxexx x sin103cos98 23

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SummarySummary

Have we met our learning objectives?Have we met our learning objectives?Specifically:Specifically: are you able to: are you able to:

• Define integration in terms of the Define integration in terms of the anti-derivative anti-derivative

• Integrate Integrate simplesimple algebraic functions algebraic functions• Use the fundamental theorem of Use the fundamental theorem of

integral calculusintegral calculus• Use math software to solve simple Use math software to solve simple

integration problems integration problems

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HomeworkHomework

1.1. Integrate:Integrate:

2. Find the area under a sine 2. Find the area under a sine wave the curve over the wave the curve over the domain x=[0,domain x=[0,] ]

3. Find the area between the 3. Find the area between the curvescurves

x x x3 22 5 6

xy 42

2xy

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Examination Type Examination Type questionsquestionsGiven the following function is to represent a Given the following function is to represent a

probability densityprobability density

a)a) Explain what is meant by the term random Explain what is meant by the term random variablevariable

b)b) State the domain of State the domain of xx under these conditions under these conditionsc)c) Use integration and the continuity of Use integration and the continuity of

probability to evaluate probability to evaluate k.k.d)d) Show that all measures of the central Show that all measures of the central

tendency are equaltendency are equale)e) Evaluate the variance and hence the standard Evaluate the variance and hence the standard

deviation of the resulting distributiondeviation of the resulting distribution

)16()( xkxxp

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Examination type Examination type questionsquestions

2. 2. Integrate the following:Integrate the following:

a)a)

b)b)

c)c)

3. 3. Find via integration the area Find via integration the area bonded by the curves and in the bonded by the curves and in the domain x=[1,2]domain x=[1,2]

)3sin(8 x

xx sin63cos12 xex 42 4/2

and )2exp(3 xy xey 2

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Revision ExerciseRevision Exercise

All you need to do now is complete the All you need to do now is complete the REVISION SHEET REVISION SHEET and you will be and you will be ready for all the calculus questions on ready for all the calculus questions on the forthcoming end test. Solutions to the forthcoming end test. Solutions to this will be given in class or is this will be given in class or is available from tutors (referral work)available from tutors (referral work)

Congratulations you have now Congratulations you have now completed the analytical toolbox completed the analytical toolbox

coursecourse