Analytical Toolbox Integral CalculusBy Dr J.P.M. Whitty

2 Learning objectives After the session you will be able to: After the session you will be able to: Define integration in terms of the anti-derivativeDefine integration in terms of the anti-derivative Integrate simple algebraic functionsIntegrate simple algebraic functions Use the fundamental theorem of integral calculusUse the fundamental theorem of integral calculus Use math software to solve simple integration problemsUse math software to solve simple integration problems

3 Integral Calculus "It is interesting that, contrary to the customary order of presentation found in our college courses where we start with differentiation and later consider integration, the ideas of the integral calculus developed historically before those of differential calculus. Some time later, differentiation was created in connection with problems on tangents to curves and with questions about maxima and minima, and still later it was observed that integration and differentiation are related to each other as inverse operations"

4 Integration Today we view integration as the inverse of differentiation and we work from the premise of gradients of tangents to derive the equations of curves. We also try to establish an algorithm that will make it easy to work backwards. To differentiate a polynomial function f(x), we use the notation

5 Inverse of differentiation We can relate a given derivative with the corresponding f(x). This asks you to work in reverse order from differentiating problems, where you were given f(x) and asked to derive f '(x). This time, you have been given the derivative and asked to work back up the chain - what function produced this derivative?

6 Lemma: Instead of subtracting form the index we must add and instead of multiplying by the index we must divide by the new index. This leads us to the Lemma.

7 Integration of standard functions It is usual in calculus textbooks that to see tables of standard functions and their respective integrals For this introductory course you will only require the following For this introductory course you will only require the following

8 Class Examples Time Copy and complete the following table. YOU MUST ALWAYS REMEMBER THE CONSTANT OF INTEGRATION!!

9 Class examples 1. Integrate the following 2. Find the following integrals

10 Class examples solutions 1) Write as and

11 Class examples solutions cont… 2) Just find the anti-derivatives thus: and Each time we MUST REMEMBER the +c

12 Further Examples Integrate the following: a) b) c) d) e)

13 Theorem: The fundamental theorem of calculus: Basically states that the area under any curve can be found via integration and application of limits, thus: Basically states that the area under any curve can be found via integration and application of limits, thus:

14 Area of a triangle We will illustrate this theorem via a elementary example. Consider a 45 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

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

16 Example Find the area under the curve the x axis and when x=1 and x=3 Square units

17 Area between curves The theorem is useful when deteriming the areas between two curves. Here we simply subtract the larger integral from the smaller one. For example. Find the area between the two curves: The only trick here is to find where the curves cross in order to find the limits of integration

18 Solution: Solve the problem simultaneously to find the limits of integration: Now find the integrals: The result is of course the big area minus the little one: Square Units

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

20 MATLAB: Integration The process is the same as usual. i.e. easy as ABC! 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.

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

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

23 Summary Have we met our learning objectives? Specifically: are you able to: Define integration in terms of the anti- derivativeDefine integration in terms of the anti- derivative Integrate simple algebraic functionsIntegrate simple algebraic functions Use the fundamental theorem of integral calculusUse the fundamental theorem of integral calculus Use math software to solve simple integration problemsUse math software to solve simple integration problems

24 Homework 1. Integrate: 2. Find the area under a sine wave the curve over the domain x=[0,  ] 3. Find the area between the curves

25 Examination Type questions Given the following function is to represent a probability density a)Explain what is meant by the term random variable b)State the domain of x under these conditions c)Use integration and the continuity of probability to evaluate k. d)Show that all measures of the central tendency are equal e)Evaluate the variance and hence the standard deviation of the resulting distribution

26 Examination type questions 2. Integrate the following: a) b) b) c) c) 3. Find via integration the area bonded by the curves and in the domain x=[1,2] and

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