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Tutoring Integral Calculus concepts needed to help others learn S Gill Williamson ©S Gill Williamson 2012 Preface From 1965 to 1991 I was a professor in the Department of Mathematics at the University of California, San Diego (UCSD) I taught many calculus classes large and small during this period In 1991 I transferred to the Department of Computer Science and Engineering - my calculus teaching days were over Recently (2011), cleaning out some files, I came across a long lost typewritten handout that I used to give to students who officially or unofficially wanted to tutor for my integral calculus classes I had fun rereading this “tutors’ guide” so I decided to redo it in LaTeX and bring it up to date with respect to online resources now regularly used by students This material assumes that as a prospective integral calculus tutor you have mastered the standard undergraduate level differential and integral calculus courses The most common conceptual and pedagogical pitfalls of tutoring integral calculus are discussed along with worked exercises S Gill Williamson, 2012 http : \www.cse.ucsd.edu\ ∼ gill Table of Contents Chapter 1: Integrals as Antiderivatives 1.1: Integrals as the inverse of differentiation 1.2: Properties of the integral 1.2.2: Chain rule in reverse 10 1.2.4: Differential notation 11 1.3: Exercises (Integration by substitution) 13 1.4: Using computer resources .15 1.5: Additional techniques of integration 18 1.6: Exercises (Integration by parts) 19 1.6.5: Integral tables and online math resources 20 Chapter 2: Fundamental Theorem and Definite Integrals 2.1: The fundamental theorem of calculus 23 2.2: Definite integrals and Riemann sums 28 2.2.1: Definite integrals 28 2.2.2: Riemann sums 29 2.3: Exercises (Areas bounded by curves) 31 2.4: Inverse trig functions, tables and index 36 TABLE OF INTEGRALS 40 TRIGONOMETRIC FUNCTIONS 63 INDEX 67 Chapter Integrals as Antiderivatives 1.1 Integration as the inverse of differentiation This material assumes that you have had a course in calculus through integral calculus and want to tutor (or home tutor) students who are studying integral calculus for the first time We focus on developing the skill and intuition to prepare yourself to face a small group of students who are stuck or, worse yet, not stuck and full of questions We start with a review of basics In our discussion, we have in mind real valued functions defined for all real numbers or intervals of real numbers just like the ones you studied in calculus up to this point d d The symbol “ dx " is used to mean “the derivative of." Thus, dx sin(x ) means “the derivative of sin(x )." Correspondingly, we use the symbol “ " to mean “the integral of" or “antiderivative of.” Thus, we write 2x cos(x ) = sin(x ) d to mean the integral of 2x cos(x ) is sin(x ): dx sin(x ) = 2x cos(x ) (1.1.1) Figure : Integral vs Derivative If f (t ) is specified as a function of t then f (t ) is commonly understood to d be a function F (t ) such that dt F (t ) = f (t ) But what if we walk into an abandoned classroom and see “ =?" on the blackboard What was the variable? If it was x, then the answer is 2x If the variable was t, then the answer is 2t To avoid this and related confusions, the notation for integrals or antiderivatives is written f (x ) dx or f (t ) dt Thus, if we had seen “ dt =?" then the answer would have been 2t If we had seen “ dx =?" then the answer would have been 2x F (x ) and F (x ) + C have the same derivative There is another simple but important observation about integrals As we d have noted, 2x cos(x ) = sin(x ), which means that dx sin(x ) = 2x cos(x ) d d (sin(x ) + 10) = 2x cos(x ) also In fact, dx (sin(x ) + C) = But, of course, dx 2x cos(x ) for any constant function C This fact is sometimes incorporated into the notation for integrals by writing (1.1.2) 2x cos(x ) dx = sin(x ) + C This notation is intended to remind us that there are infinitely many functions with derivative 2x cos(x ) and they all differ by a constant function Once this observation has been made and we understand what we are talking about, it is quite all right to write simply 2x cos(x )dx = sin(x ) We understand in this latter notation that sin(x ) is a representative from an infinite class of antiderivatives for 2x cos(x ), and all of the rest are obtained by adding a constant function to sin(x ) H (x ) = =⇒ H (x ) = C , a constant function The fact that all antiderivatives of a given function f (x ) differ by a constant is a subtle idea Suppose we have two functions, F (x ) and G (x ), such that F (x ) = G (x ) = f (x ) Let H (x ) = F (x ) −G (x ) Then H (x ) = F (x ) −G (x ) = f (x ) − f (x ) = To claim that F (x ) and G (x ) differ by a constant function is the same as claiming that H (x ) = F (x ) − G (x ) is a constant function This means that the statement that “any two antiderivatives F (x ) and G (x ) of f (x ) differ by a constant" is the same as the statement that “any function H (x ) with derivative function the zero function must be a constant function." This latter statement has strong intuitive appeal Suppose H (x ) = for all x Let’s try to draw the graph of such an H (x ) Suppose H (0) = 2, for example Put your pencil at the point (0, 2) and try to imagine what the graph is like near this point If, in going right or left, you draw the graph with the slightest bit of slope up or down you will construct points on the graph where H (x ) is not You’re stuck at H (x ) = and must draw the graph of the constant function For more advanced courses in mathematical analysis it is essential that this intuitive idea be given a precise analytical formulation Constant Functions Can Wear Many Disguises Here is another complication Suppose John decides that sin(2x ) dx = − cos(2x ) and suppose that Mary decides that sin(2x )dx = sin2 (x ) If they are both right (and they are in this case) then sin2 (x ) and − cos(2x ) must differ by a constant (i.e., sin2 (x ) − (− cos(2x )) = sin2 (x ) + cos(2x ) is a constant function) If you know your basic trigonometric identities, then you will recognize that this is true and, in fact, sin2 (x ) + cos(2x ) = Thus, just because two integrals F (x ) and G (x ) for f (x ) must differ by a constant doesn’t mean that they are easily recognizable as differing by a constant 1.2 Properties of the integral The most basic property of integrals is “linearity." This property, which we have already used, is stated in Theorem 1.2.1 Theorem 1.2.1 Let f (x ) and д(x ) be functions and α and β numbers Then (αf (x ) + βд(x ))dx = α f (x )dx + β д(x )dx Proof This follows directly from the definition of the integral together with d linearity of dx Let F (x ) and G (x ) be the antiderivatives of f (x ) and д(x ) d Then dx (αF (x ) + βG (x )) = αf (x ) + βд(x ), which means, by definition of the integral, that (αf (x ) + βд(x ))dx = αF (x ) + βG (x ) Substituting F (x ) = f (x ) dx and G (x ) = д(x ) dx gives the result Your students’ main task in computing antiderivatives or integrals will be to develop systematic ways to reduce new problems to ones they have already solved Theorem 1.2.1 is a start in this direction We have already noticed that 2x cos(x ) dx = sin(x ) and sin(2x ) = sin2 (x ) Thus we can evaluate an integral such as (2πx cos(x ) + 25 sin(2x )) dx by using Theorem 1.2.1: 2x cos(x ) dx + 25 π sin(2x ) dx = π sin(x ) + 25 sin2 (x ) Your students already know many integrals As they begin to compute more integrals, your students will use the rule of Theorem 1.2.1 automatically They will discover that they already know many integrals! Every differentiation formula they have memorized gives rise to a corresponding integration formula: d sin(x ) = cos(x ) becomes sin(x ) = cos(x )dx dx d ln(|x |) = 1/x becomes ln |x | = dx dx x d n nx n−1dx x = nx n−1 becomes x n = dx This latter integral has been memorized by millions of calculus students as x ndx = For n = −1, Show: x n+1 n+1 for n −1 x −1dx = ln |x | d dx sec(x ) = sec(x ) tan(x ) d dx csc(x ) = − csc(x ) cot(x ) By applying Theorem 1.2.1, you can compute integrals such as (34 sin(x ) + 23x + 45 sec2 (x ))dx = −34 cos(x ) + 23 (1.2.2) x4 + 45 tan(x ) Chain rule in reverse To really get started on the problem of computing integrals, your students must learn how to the chain rule in reverse In particular, (1.2.3) d f (д(x )) = f (д(x ))д (x ) =⇒ f (д(x )) = dx 10 f (д(x ))д (x )dx 56 57 58 59 60 61 62 63 64 rl (no) d( u! u\ dr _r l u, I dr dn'df r tl (ttu\ -: dr' ):z \ - sr (1 + log, r) u _ ?rlr, dr dx ti Tt+ afr dfr r/ sin c : at du dn cl cos r , dtt uT ar , (;) : =- dn ,u2 cJu, T ur sln r dn r l r : t t r, r : -dt - (:sc,'r' r/ sec r , tittt;r.secrt (t, : TY ar t - t, c/ csc r ,/" de' :-etrtr.(:scr gr 7: d,fr r/ silr- l.r -5-e du rlu" : o" ,fr Logez', ,I* d eos- lr -a eOS.tr, ,/ t;ur r; (l',f' clt( d1f (tt) gf ,dJtt * + drz du dnt ' du,z d*'' drn 3J ff:se(;2.r rl,{(u\_ d,f(u,).4y, cln g ,r,r _l _ & Iog s dn rlts drt dr 1( l o g , , r ) _ I eiz d cschc - rr Vl de d eosh-r tr _ _ ,% - d ettr- l.n t t - rV e i-l dr rV rt- d tanh-r n dfr 1 -n2 _o dn t *n d sesh-r c id sinh r de ?:cosht r -1 n\E -n2 d eoslrr : sinh c d d csch-r s d tanlr r *"f"',f(4n*- de d sech ,t : dn, -1 %a da -f f: seehtr d ctnh c r \6r -L d ctnh-rc des c - lr V r r }l - -E -_ l dn : dn I *rt d see,-I r a r-c a ,t" : - cscho ,ctnh s I + ,'' -. cln Vi-= d sinh- r e' dr rf krtr-rtr - I clc *Ir@)dr csclrsc - sech s -tanh fr 65 n {n'+ f (b) _ - f(o) 66 Index antiderivative, arcsin or sin −1 , 16 area actual, 31 integration by parts, 18 integration limits of, 29 linearity, chain rule in reverse, 10 computer resources, 15 contents, parameter, 31 parameterized problems, 31 Riemann sum definition, 29 Riemann sum discussion, 29 differential chain rule, 12 definition, 12 notation, 11 signed area definition, 25 discussion, 24 exercises, 31 summary, 28 tutoring points, 27 substitution exercises, 13 substitution method, 12 envelope game, 27 fundamental theorem calculus, 25 integral, additive constant, integral definite, 28 integral parameterized, 20 integral recursive, 20 integral summary, 28 integrals definite, 60 fundamental forms, 40 irrational algebraic involving (a + bx ) 1/2 , 44 involving (a+bx+cx ) 1/2 , 47 involving (x ±a ) 1/2 , 45 rational algebraic involving (a + bx ), 41 involving (a+bx+cx ), 43 involving (a + bx n ), 42 transcendental, 48 same but not obvious, table of integrals, 40 trig functions cos and arccos, 37 sin and arcsin, 37 tan and arctan, 38 cot and arccot, 39 csc and arccsc, 38 sec and arcsec, 39 inverse, 36 trig identites, 62 variations 67 sec(x ) dx, 17 NOTES 68 NOTES 69 NOTES 70