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MAT 136 Calculus I Lecture Notes
Li Chen
November 15, 2018
Contents
1 How to Use These Notes 1
2 Introduction to the Integral 2
2.1 (5.1) Area and Distances . . . . . . . . . . . . . . . . . . . . . . .
2
2.1.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1.2 Displacement/Distance . . . . . . . . . . . . . . . . . . .
3
2.1.3 Sigma Notation . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2 (5.2) The Denite Integral . . . . . . . . . . . . . . . . . . . . . .
4
2.2.1 Denition of the Denite Integral . . . . . . . . . . . . . .
4
2.2.2 Properties of the Sigma Sum . . . . . . . . . . . . . . . .
5
2.2.3 Properties of the Integral . . . . . . . . . . . . . . . . . .
5
2.3 (4.9) Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4 (5.3) Fundamental Theorem of Calculus (FTC) . . . . . . . . . .
6
2.5 (5.4) Indenite Integrals . . . . . . . . . . . . . . . . . . . . . . .
7
3 Techniques of Integration 9
3.1 (5.2) Linearity [? ? ? ? ?] . . . . . . . . . . . . . . . . . . . . . . .9
3.2 (5.5) The Substitution Rule . . . . . . . . . . . . . . . . . . . . .
10
3.3 (7.1) Integration by Parts . . . . . . . . . . . . . . . . . . . . . .
14
3.4 (7.1) Trig Integrals . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4 Trig Substitution 19
5 Other Techniques 19
1 How to Use These Notes
First let me state the two most important questions we want to answer in this class
How to integrate a functionf(x)
1 The majority of the contents in this note is geared to answer this questions. In these notes, I will explicitly indicate which are the important materials. Although one should be comfortable with all the materials in this note, but certain critical concept must be emphasized to illuminate the essence of the course. First, denitions are inbold. I will use a scale of stars to measure the importance of the material. The importance of the concept decreases as the number of star decreases. Whenever you see a 5-star tag [? ? ? ? ?], this means you must know this inside out. And [?] is just a bit more important than the regular text you see in the notes. Finally, I willitalicizeitem that 1) you need to be careful with as they might be confusing, or 2) to leave a comment. The corresponding section numbers in [Stewart] are listed in bracket in the heading of each section.And I will NOT write the integration constant+Cin the notes to save some ink, but you should always write it!!
2 Introduction to the Integral
2.1 (5.1) Area and Distances
2.1.1 Area
A mathematical illustrative example of the integral is area under a curve. Let f(x) be anon-negativecontinuous function. We will say thearea undery= f(x)fromx=atox=bto be the area bounded between the linesx=a, x=b, thex-axis, and the graph off(x).x432101234y
3210123
f(x) =x22 In this example, the shaded region represents the area under the curvey= f(x) =x2fromx=2 tox= 2. In general, to nd the area under the curve y=f(x) fromx=atox=b, we divide the interval [a;b] into segments [x0;x1];[x1;x2];;[xn1;xn] (2.1.1) of width x. That is, x=xixi1=ban (2.1.2) andxi=a+ixfor alli= 1;:::;n. On each interval, the funtionf(x)f(xi) since the interval is very small so thatf(x) is about constant. So that the area underf(x) is approximatelyf(xi)x. Hence, we can then approximate the area by f(x1)x+f(x2)x++f(xn)x:(2.1.3) In fact the choice ofsample pointdoes not matter. In fact, we can choose any sample pointxiin the interval [xi1;xi] and compute the sum f(x1)x+f(x2)x++f(xn)x:(2.1.4) As we choose the interval [xi1;xi] to be small, equivalently xto be smaller. As we choose the the width xto smaller by taking the number of segments n! 1, thearea underf(x)fromx=atox=bis
A= limn!1f(x1)x+f(x2)x++f(xn)x:(2.1.5)
2.1.2 Displacement/Distance
A physically motivating example for the integral is the displacement traveled by a car with velocityf(t) at timet. Suppose that from timet=atot=ba car travels at a velocityf(t). Iff(t) =vis a constant. Then the displacement traveled in tunits of time is simply d=vt=f(t)t:(2.1.6) Now suppose thatf(t) is variable. If it is continuous, then its value varies slightly iftchanges by a small amount. Hence we may approximate the distance traveled by dividing the time interval [a;b] into small segments [t0;t1];[t1;t2];;[tn1;tn] (2.1.7) of width t=ban . On each time segment [ti1;ti], the car travels approxi- mately f(ti)t(2.1.8) 3 units of length, wheretiis any sample point in the interval [ti1;ii]. The total displacement,d, is, therefore, approximately, df(t1)t+f(t2) ++f(tn)t:(2.1.9) Taking the segments [ti1;ti] smaller for better approximation, we see that d= limn!1f(t1)t+f(t2) ++f(tn)t:(2.1.10)
2.1.3 Sigma Notation
We will henceforth denote use thesigma notation
n X i=1a i=a1+a2++an:(2.1.11)
Then the area under the curvef(x) fromx=atox=bis
A= limn!1n
X i=1f(xi)x[? ? ?] (2.1.12) and the displacement traveled in the time interval [a;b] by a car of velocityf(t) is d= limn!1n X i=1f(ti)t[? ? ?] (2.1.13)
2.2 (5.2) The Denite Integral
2.2.1 Denition of the Denite Integral
Letf(x) be a function dened on [a;b]. We partition [a;b] in tonintervals of equal width x=ban [x0;x1];[x1;x2];;[xn1;xn] (2.2.1) wherea=x0andb=xn. We dene theintegral off(x)fromatobto be Z b a f(x)dx= limn!1n X i=1f(xi)x(2.2.2) provided this limit exists and is independent of the choice of sample pointsxi.
If the limist exists, we say thatfis integrable.
The following names are given to the parts of the integral Z b a |{z} integral signf(x)|{z} integranddx |{z} integrate with respect tox(2.2.3) 4 We remark that the symboldxonlyindicate that we are integrating with respect tox, should there be any more variables appearing in the integrand.
The sum
n X i=1f(xi)x(2.2.4) is called aRiemann sum.
2.2.2 Properties of the Sigma Sum
The following list contains properties of the sigma sum. The rst three properies are the most important. The rest are useful when we compute integrals explicitly from its denition. (The rst three are important. Do not memorize the last
4 properties, they can be readily searched on Google and will be provided in a
formula sheet for any tests if a question demands it) 1. Pn i=1cai=cPn i=1ai. 2. Pn i=1(ai+bi) =Pn i=1ai+Pn i=1bi. 3.
If aibifor alli, thenPn
i=1aiPn i=1bi. 4. Pn i=11 =n. 5. Pn i=1i=n(n+1)2 6. Pn i=1i2=n(n+1)(2n+1)6 7. Pn i=1i3=n(n+1)2 2
2.2.3 Properties of the Integral
The rst three properties of the sigma sum translates, through the Riemann sum, into properties for the integral. The last two properties listed below does not come from a property listed in the previous subsection. 1. [ ? ? ?]Rb acf(x)dx=cRb af(x)dx(linearity) 2. [ ? ? ?]Rb af(x) +g(x)dx=Rb af(x) +Rb ag(x)dx(linearity) 3. Rb a1dx=ba 4.
If f(x)g(x), thenRb
af(x)dxRb ag(x) 5. [ ??] Ifabc, thenRc af(x)dx=Rb af(x)dx+Rc bf(x)dx. 6. [ ??]Rb af(x)dx=Ra bf(x)dx. 5
2.3 (4.9) Antiderivatives
First, I want to dispel any possible confusion in notation.For our purpose, a real function is a map from an intervalIto the real line. I will usebothfand f(x) to mean such a function. However, confusion may arise when we want to talk about a function evaluated at a pointx=x0. In this case the symbolf(x0) would mean the functionfevaluated at the pointx=x0. In the former case, the argumentxinf(x) is generic, while in the latter case the pointx0is given and xed. In what follows, I will usefto denote a function. But the meaning off(x), either as a function or as the evaluation offat the pointx, depends on the context. If needed, I will specify its meaning. Given any functionf(x) on [a;b], anantiderivativesis any functionF(x) on [a;b] such that F
0(x) =f(x):(2.3.1)
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[PDF] Techniques of Integration