DOCSLIB.ORG

Integral Calculus by Jorge Samayoa

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integral Calculus by Jorge Samayoa Integral Calculus By Jorge Samayoa §1. (Short)History The history of calculus stares several hundreds of years ago, when the mathematician Archimedes (225 B.C.) made a very important contribution. It said that the area of a 4 2 segment of a parabola is the area of a triangle with the same base and vertex and 3 3 the area of the circumscribed parallelogram. In his work, Archimedes constructed an infinite series of triangles of known area to approximate the area of such segment. Archimedes, therefore, introduces the concept of putting together different pieces of known area to get a different area. This seems to be a very basic thought, but it was very influential to the basis of calculus. Several years after Archimedes work, and several contributions of different people, there were two people that took, pretty much, the whole credit of what is know as calculus. These people were, Sir Isaac Newton and Gottfried Leibniz. Even thought the first person who gives a precise idea of calculus was Newton, the one that publish first his work know as calculus was Leibniz. At that time, there was a controversy of who invented calculus first and which country deserved the credit. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions". Nowadays, independently of whose ideas we should follow, calculus is divided in two huge problems: The first problem of calculus, which is the estimation of the slope of a curve in a given point, and the second problem of calculus which is the estimation of the area under a curve in a given interval. In this document, however, we will focus in the second problem of calculus since we already went over the first problems in the last tutorial (The Geometric Approach of the Derivative). §2. The Second Problem of Calculus (The Area Problem) (http://calclab.math.tamu.edu/~jsamayoa/final_section_2 ) The second problem of calculus is finding the area of the region R that lies under the curve y = fx() from a to b (Figure 1). Figure 1: Area under the curve In order to find the area under a curve, as described in the section below, we will approximate the area by using another figure of known area; the rectangle (Figure 2). Figure 2: Area of the Rectangle One of the persons who introduced this was Jean-Gaston Darboux. However, this approach is equivalent to the approach given by Friedrich Bernhard Riemann. There are two different ways of doing this approximation. One is the Darboux upper sums, figure 3, and the Darboux lower sums, figure 4. As you might notice, the difference relies on the selection of the lengths of the approximating rectangles. Let fab:[ , ]→ be a bounded function, and let Px= (0 ,… , xn ) be a partition of [ ab , ]. Let Mfxi = sup ( ). xx∈[,]ii−1 x The upper Darboux sum of f with respect of P is n U[fP , ]=Δ∑ Mii x . i=1 where Δ=xiii(xx −−1 ). Figure 3: Upper Darboux sum. Let f :[ab , ]→ be a bounded function, and let Px= (0 ,… , xn ) be a partition of [ ab , ]. Let mfxi = inf ( ). xx∈[,]ii−1 x The lower Darboux sum of f with respect of P is n LfP[,]=Δ∑ mii x . i=1 where Δ=xxxiii( −−1 ). Figure 4: Lower Darboux sum. Theorem 1: Let AyfxabP be the area of the region R that lies under the curve = ( ) from to , and let be a Partition of [ab , ]. Then, LfP[ , ]≤≤ A UfP [ , ] Proof: (We let the proof of this result to the reader). In order to improve our approximation of the area A. Let us make a refinement of P0 in the Darboux upper sums, say P1. Notice that the difference between the approximation A using P0 and using P1 is the length of the interval[,xiix −1 ]. If we refine Pi by Pj (i<j) the approximation will be better and hxx= ii−=Δ−1 x i is getting smaller while n is getting bigger (figure 5). Figure 5: As hn→→∞0 as . This led us to the following definitions: Definition 1: The area A of the region R that lies under the graph of f ∈Cab[,] is the limit of the Darboux sums : AUfPLfP= lim [ , ]= lim [ , ] n→∞ n→∞ where Px= (,0 … , xn )is a partition of [ab , ]. Equivalently, we state de following definition. Definition 2: th Let Ai be the area of the i approximating rectangle. Then, the area A of the region R that lies under the graph of f ∈Cab[ , ] is A==lim Ainn lim[ fx (11 ) Δ+Δ++Δ x fx ( 2 ) x 2 fx ( ) x ] nn→∞ →∞ for x j ∈[,]ab. Example 1: Estimate the area under the curve y = x2 for x∈[0,1] by 1. 4 approximating rectangles by 1.1. Darboux upper sums. 1.2. Darboux lower sums. 2. n approximating rectangles, then, let n →∞. SOLUTION: 1.1 ba− 1 Δ=x = n 4 ⎛⎞113 ⎛⎞ ⎛⎞ Af≈ ⎜⎟Δ+ xf ⎜⎟ Δ+ xf ⎜⎟ Δ+ xf()1 Δ x ⎝⎠424 ⎝⎠ ⎝⎠ 1119⎛⎞ ≈+++=⎜⎟1 0.4688 4164⎝⎠ 16 1.2 ba− 1 Δ=x = n 4 ⎛⎞113 ⎛⎞ ⎛⎞ Af≈ ()0 Δ+ xf⎜⎟ Δ+ xf ⎜⎟ Δ+ xf ⎜⎟ Δ x ⎝⎠424 ⎝⎠ ⎝⎠ 1119⎛⎞ ≈+++=⎜⎟0 0.2188 416416⎝⎠ 2 ba− 1 Δ=x = nn ⎛⎞12 ⎛⎞ Af≈Δ+Δ++Δ⎜⎟ xf ⎜⎟ x f()1 x ⎝⎠nn ⎝⎠ 22 2 11⎛⎞⎛⎞ ⎛⎞ 2 ⎛⎞n ≈+++ ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ nn⎝⎠⎝⎠ ⎝⎠ n ⎝⎠ n 11(1)(21)22 2 ⎛ nn++ n ⎞ ≈+++=22()12 n ⎜ nn⎝ 6 ⎠ ⎛⎞(1)(21)1nn++ ∴ A ==lim⎜⎟ . n→∞ ⎝⎠63n §3. Definite Integral. (http://calclab.math.tamu.edu/~jsamayoa/final_section_3 ) Definition 3: Suppose f ∈Cab[,].Let us divide the interval [,]ab in n subintervals of width ba− Δ=x . Let x ==ax,,, x… , x b, be the endpoints and x***,,,,xx… x *the sample n 012n 01 2 n * points in such intervals, so xiii lies in the ixxth subinterval [−1 , ]. Then the definite integral of f from a to b is b n * f ()xdx= lim f xi Δ x ∫ n→∞ ∑ () a i=1 Since we supposed that f is continuous in [a , b], definition 3 can be states as follows. For every numberε > 0 , there is a positive integer number N such that, b n fxdx() − f x* Δ< x ε ∫ ∑ ()i a i=1 * For every n > N and for every xiiiin [xx−1 , ]. Example 2: 1 Evaluate ∫ xdx. 0 SOLUTION: 1 n * By definition f (xdx )=Δ lim f xi x . ∫ n→∞ ∑ () 0 i=1 ba− 1 Now, the width is Δ=x = . nn thus, 1 ⎛⎞12 ⎛⎞ xdx≈Δ+Δ++Δ f⎜⎟ x f ⎜⎟ x f()1 x ∫ ⎝⎠nn ⎝⎠ 0 11⎛⎞⎛⎞⎛⎞ 2 ⎛⎞n ≈+++⎜⎟⎜⎟⎜⎟ ⎜⎟ nn⎝⎠⎝⎠⎝⎠ n ⎝⎠ n 11(1)⎛⎞nn+ ≈+++=22()1 2 n ⎜⎟ nn⎝⎠2 1 ⎛⎞(1)1n + ∴=xdx lim = . ∫ n→∞ ⎜⎟ 0 ⎝⎠22n §3.1. Properties of the definite integral. Due the nature of this document, we will state some of the most important properties of the definite integral, and the proofs of such properties are lived as exercise for the student. Properties of the Integral. b 1. ∫ cdx=− c( b a ), where c is any constant. a bbb 2. ∫∫∫f ()x±= gxdx () f () xdx ± gxdx () aaa bb 3. ∫∫cf( x ) dx= c f ( x ) dx , where c is any constant. aa bc c 4. ∫∫∫f ()xdx+= f () xdx f () xdx with cab∈( , ) . ab a 5. If mfxM≤≤( ) for axb ≤≤, then b mb()−≤ a∫ f ()(). xdx ≤ M b − a a Example 3: 1 Use properties of the integral to evaluate ∫ (2+ 4x )dx . 0 1111 ∫∫∫∫(2+=+=−+ 4x )dx 2 dx 4 xdx 2(1 0) 4 xdx 0000 by example 2, 1 ⎛⎞1 ∫ (2+=+= 4xdx ) 2 4⎜⎟ 4 0 ⎝⎠2 Definition 4 (Fundamental Theorem of Calculus) Supposef ∈Cab [ , ] . Then, x 1. If gx( )==∫ ftdt ( ) , then g'( x ) f ( x ). a b 2. ∫ f (xdx )=− Fb ( ) Fa ( ), where F is any antiderivative of f, that is, F' = f . a Proof: Since this is a support document for out tutorial, the proof of this will be in other document with different goal. §4. Indefinite Integrals. (http://calclab.math.tamu.edu/~jsamayoa/final_section_4 ) The goal of this section is to familiarize the student with the evaluation of indefinite integrals. Therefore, we will just present a table with several indefinite integrals so that the student might have a comprehension of the computational procedure of solving indefinite integrals. Because of the relation given by the Fundamental Theorem of Calculus between antiderivatives and integrals, the notation ∫ f (xdx ) is traditionally used for an antiderivatives of f and is called indefinite integral. Thus, ∫ f ()xdx== Fx () means F'() x f () x xdx22⎛⎞ For example, we can write ∫ xdx= ++= C, because ⎜⎟ C x . 22dx ⎝⎠ Table of Integrals. Example 4: x5 a) ()3cos3x4 +=++xdx sin x C ∫ 5 ⎛⎞2 2 b) ⎜⎟tanx +=++dx sec x 2ln x C ∫ ⎝⎠x ⎛⎞4 c) edeCθθ+=++θθ4sin−1 ∫ ⎜⎟2 ⎝⎠1−θ d) Use the Fundamental Theorem of Calculus, to 2 ⎛⎞2 evaluate 23x4 +−xdx . ∫ ⎜⎟2 0 ⎝⎠1+ x 2 2 ⎛⎞2 xx52 23xx41+− dx = 2 + 3 − 2tan− x ∫ ⎜⎟2 0 ⎝⎠152+ x 0 52 ⎛⎞22 −1 = ⎜⎟2+− 3 2 tan 2 − 0 ⎝⎠52 64 =+−6 2 tan−1 2 5 94 =−2 tan−1 2 5 §5 Area between two curves. (http://calclab.math.tamu.edu/~jsamayoa/final_section_5 ) In section 2 we defined and calculated areas of regions that lie under curves. In this section we will use integrals to find areas of region between two curves.
Load more

Recommended publications
  • Antiderivatives and the Area Problem
    Chapter 7 antiderivatives and the area problem Let me begin by defining the terms in the title: 1. an antiderivative of f is another function F such that F 0 = f. 2. the area problem is: ”find the area of a shape in the plane" This chapter is concerned with understanding the area problem and then solving it through the fundamental theorem of calculus(FTC). We begin by discussing antiderivatives. At first glance it is not at all obvious this has to do with the area problem. However, antiderivatives do solve a number of interesting physical problems so we ought to consider them if only for that reason. The beginning of the chapter is devoted to understanding the type of question which an antiderivative solves as well as how to perform a number of basic indefinite integrals. Once all of this is accomplished we then turn to the area problem. To understand the area problem carefully we'll need to think some about the concepts of finite sums, se- quences and limits of sequences. These concepts are quite natural and we will see that the theory for these is easily transferred from some of our earlier work. Once the limit of a sequence and a number of its basic prop- erties are established we then define area and the definite integral. Finally, the remainder of the chapter is devoted to understanding the fundamental theorem of calculus and how it is applied to solve definite integrals. I have attempted to be rigorous in this chapter, however, you should understand that there are superior treatments of integration(Riemann-Stieltjes, Lesbeque etc..) which cover a greater variety of functions in a more logically complete fashion.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Notes Chapter 4(Integration) Definition of an Antiderivative
    1 Notes Chapter 4(Integration) Definition of an Antiderivative: A function F is an antiderivative of f on an interval I if for all x in I. Representation of Antiderivatives: If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G(x) = F(x) + C, for all x in I where C is a constant. Sigma Notation: The sum of n terms a1,a2,a3,…,an is written as where I is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1. Summation Formulas: 1. 2. 3. 4. Limits of the Lower and Upper Sums: Let f be continuous and nonnegative on the interval [a,b]. The limits as n of both the lower and upper sums exist and are equal to each other. That is, where are the minimum and maximum values of f on the subinterval. Definition of the Area of a Region in the Plane: Let f be continuous and nonnegative on the interval [a,b]. The area if a region bounded by the graph of f, the x-axis and the vertical lines x=a and x=b is Area = where . Definition of a Riemann Sum: Let f be defined on the closed interval [a,b], and let be a partition of [a,b] given by a =x0<x1<x2<…<xn-1<xn=b where xi is the width of the ith subinterval.
    [Show full text]
  • 4.1 AP Calc Antiderivatives and Indefinite Integration.Notebook November 28, 2016
    4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 4.1 Antiderivatives and Indefinite Integration Learning Targets 1. Write the general solution of a differential equation 2. Use indefinite integral notation for antiderivatives 3. Use basic integration rules to find antiderivatives 4. Find a particular solution of a differential equation 5. Plot a slope field 6. Working backwards in physics Intro/Warm­up 1 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Note: I am changing up the procedure of the class! When you walk in the class, you will put a tally mark on those problems that you would like me to go over, and you will turn in your homework with your name, the section and the page of the assignment. Order of the day 2 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Vocabulary First, find a function F whose derivative is . because any other function F work? So represents the family of all antiderivatives of . The constant C is called the constant of integration. The family of functions represented by F is the general antiderivative of f, and is the general solution to the differential equation The operation of finding all solutions to the differential equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign: is read as the antiderivative of f with respect to x. Vocabulary 3 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 Nov 28­9:38 AM 4 4.1 AP Calc Antiderivatives and Indefinite Integration.notebook November 28, 2016 ∫ Use basic integration rules to find antiderivatives The Power Rule: where 1.
    [Show full text]
  • 12.4 Calculus Review
    Appendix 445 12.4 Calculus review This section is a very brief summary of Calculus skills required for reading this book. 12.4.1 Inverse function Function g is the inverse function of function f if g(f(x)) = x and f(g(y)) = y for all x and y where f(x) and g(y) exist. 1 Notation Inverse function g = f − 1 (Don’t confuse the inverse f − (x) with 1/f(x). These are different functions!) To find the inverse function, solve the equation f(x) = y. The solution g(y) is the inverse of f. For example, to find the inverse of f(x)=3+1/x, we solve the equation 1 3 + 1/x = y 1/x = y 3 x = . ⇒ − ⇒ y 3 − The inverse function of f is g(y) = 1/(y 3). − 12.4.2 Limits and continuity A function f(x) has a limit L at a point x0 if f(x) approaches L when x approaches x0. To say it more rigorously, for any ε there exists such δ that f(x) is ε-close to L when x is δ-close to x0. That is, if x x < δ then f(x) L < ε. | − 0| | − | A function f(x) has a limit L at + if f(x) approaches L when x goes to + . Rigorously, for any ε there exists such N that f∞(x) is ε-close to L when x gets beyond N∞, i.e., if x > N then f(x) L < ε. | − | Similarly, f(x) has a limit L at if for any ε there exists such N that f(x) is ε-close to L when x gets below ( N), i.e.,−∞ − if x < N then f(x) L < ε.
    [Show full text]
  • 1. Antiderivatives for Exponential Functions Recall That for F(X) = Ec⋅X, F ′(X) = C ⋅ Ec⋅X (For Any Constant C)
    1. Antiderivatives for exponential functions Recall that for f(x) = ec⋅x, f ′(x) = c ⋅ ec⋅x (for any constant c). That is, ex is its own derivative. So it makes sense that it is its own antiderivative as well! Theorem 1.1 (Antiderivatives of exponential functions). Let f(x) = ec⋅x for some 1 constant c. Then F x ec⋅c D, for any constant D, is an antiderivative of ( ) = c + f(x). 1 c⋅x ′ 1 c⋅x c⋅x Proof. Consider F (x) = c e +D. Then by the chain rule, F (x) = c⋅ c e +0 = e . So F (x) is an antiderivative of f(x). Of course, the theorem does not work for c = 0, but then we would have that f(x) = e0 = 1, which is constant. By the power rule, an antiderivative would be F (x) = x + C for some constant C. 1 2. Antiderivative for f(x) = x We have the power rule for antiderivatives, but it does not work for f(x) = x−1. 1 However, we know that the derivative of ln(x) is x . So it makes sense that the 1 antiderivative of x should be ln(x). Unfortunately, it is not. But it is close. 1 1 Theorem 2.1 (Antiderivative of f(x) = x ). Let f(x) = x . Then the antiderivatives of f(x) are of the form F (x) = ln(SxS) + C. Proof. Notice that ln(x) for x > 0 F (x) = ln(SxS) = . ln(−x) for x < 0 ′ 1 For x > 0, we have [ln(x)] = x .
    [Show full text]
  • Antiderivatives Math 121 Calculus II D Joyce, Spring 2013
    Antiderivatives Math 121 Calculus II D Joyce, Spring 2013 Antiderivatives and the constant of integration. We'll start out this semester talking about antiderivatives. If the derivative of a function F isf, that is, F 0 = f, then we say F is an antiderivative of f. Of course, antiderivatives are important in solving problems when you know a derivative but not the function, but we'll soon see that lots of questions involving areas, volumes, and other things also come down to finding antiderivatives. That connection we'll see soon when we study the Fundamental Theorem of Calculus (FTC). For now, we'll just stick to the basic concept of antiderivatives. We can find antiderivatives of polynomials pretty easily. Suppose that we want to find an antiderivative F (x) of the polynomial f(x) = 4x3 + 5x2 − 3x + 8: We can find one by finding an antiderivative of each term and adding the results together. Since the derivative of x4 is 4x3, therefore an antiderivative of 4x3 is x4. It's not much harder to find an antiderivative of 5x2. Since the derivative of x3 is 3x2, an antiderivative of 5x2 is 5 3 3 x . Continue on and soon you see that an antiderivative of f(x) is 4 5 3 3 2 F (x) = x + 3 x − 2 x + 8x: There are, however, other antiderivatives of f(x). Since the derivative of a constant is 0, 4 5 3 3 2 we can add any constant to F (x) to find another antiderivative. Thus, x + 3 x − 2 x +8x+7 4 5 3 3 2 is another antiderivative of f(x).
    [Show full text]
  • Derivatives and Antiderivatives
    Lesson R MA 16020 Nick Egbert Note. There should be no new information in this lesson. This is a brief review of things you should have learned in Calculus I, but certainly not exhaustive. Derivatives and antiderivatives There are several derivative anti derivative rules that you should have pretty well-memorized at this point: It is very important that you know these well to make the transition into this course go smoothly. Definition. Given a continuous function f(x), if F 0(x) = f(x); we say that F (x) is an antiderivative of f(x); and we write Z F (x) = f(x) dx: 1 Lesson R MA 16020 Nick Egbert Remark. If F (x) is an antiderivative of f(x) then the indefinite integral of f is given by Z f(x) dx = F (x) + C; where C is some constant. This means that any two functions which differ by only a constant will have the same antiderivative. Remark. We want to note a couple things about the notation R f(x)dx: 1. The process of computing R f(x)dx can be called antidifferentiation 5 integration 5 taking or evaluating the integral 2. f(5x) is called the integrand. 3. Writing the \dx" on the outside is essential. This tells us that we are integrating with respect to x. It will especially be important in MA 16020 when we start having integrals with more than one variable. Fundamental theorem of calculus Theorem. If f is a real-valued continuous function on the interval [a; b]; and F the antiderivative of f: Z x F (x) = f(t) dt; a Then F 0(x) = f(x) for all x in the interval (a; b): In particular, we have that Z b f(t) dt = F (b) − F (a): a This has an important geometric interpretation.
    [Show full text]
  • Multiple Integrals
    Chapter 11 Multiple Integrals 11.1 Double Riemann Sums and Double Integrals over Rectangles Motivating Questions In this section, we strive to understand the ideas generated by the following important questions: What is a double Riemann sum? • How is the double integral of a continuous function f = f(x; y) defined? • What are two things the double integral of a function can tell us? • Introduction In single-variable calculus, recall that we approximated the area under the graph of a positive function f on an interval [a; b] by adding areas of rectangles whose heights are determined by the curve. The general process involved subdividing the interval [a; b] into smaller subintervals, constructing rectangles on each of these smaller intervals to approximate the region under the curve on that subinterval, then summing the areas of these rectangles to approximate the area under the curve. We will extend this process in this section to its three-dimensional analogs, double Riemann sums and double integrals over rectangles. Preview Activity 11.1. In this activity we introduce the concept of a double Riemann sum. (a) Review the concept of the Riemann sum from single-variable calculus. Then, explain how R b we define the definite integral a f(x) dx of a continuous function of a single variable x on an interval [a; b]. Include a sketch of a continuous function on an interval [a; b] with appropriate labeling in order to illustrate your definition. (b) In our upcoming study of integral calculus for multivariable functions, we will first extend 181 182 11.1.
    [Show full text]
  • Appendix: TABLE of INTEGRALS
    -~.I--­ Appendix: TABLE OF INTEGRALS 148. In the following table, the constant of integration, C, is omitted but should be added to the result of every integration. The letter x represents any variable; u represents any function of x; the remaining letters represent arbitrary constants, unless otherwise indicated; all angles are in radians. Unless otherwise mentioned, loge u = log u. Expressions Containing ax + b 1. I(ax+ b)ndx= 1 (ax+ b)n+l, n=l=-1. a(n + 1) dx 1 2. = -loge<ax + b). Iax+ b a 3 I dx =_ 1 • (ax+b)2 a(ax+b)· 4 I dx =_ 1 • (ax + b)3 2a(ax + b)2· 5. Ix(ax + b)n dx = 1 (ax + b)n+2 a2 (n + 2) b ---,,---(ax + b)n+l n =1= -1, -2. a2 (n + 1) , xdx x b 6. = - - -2 log(ax + b). I ax+ b a a 7 I xdx = b 1 I ( b) • (ax + b)2 a2(ax + b) + -;}i og ax + . 369 370 • Appendix: Table of Integrals 8 I xdx = b _ 1 • (ax + b)3 2a2(ax + b)2 a2(ax + b) . 1 (ax+ b)n+3 (ax+ b)n+2 9. x2(ax + b)n dx = - - 2b--'------'-- I a2 n + 3 n + 2 2 b (ax + b) n+ 1 ) + , ni=-I,-2,-3. n+l 2 10. I x dx = ~ (.!..(ax + b)2 - 2b(ax + b) + b2 10g(ax + b)). ax+ b a 2 2 2 11. I x dx 2 =~(ax+ b) - 2blog(ax+ b) _ b ). (ax + b) a ax + b 2 2 12 I x dx = _1(10 (ax + b) + 2b _ b ) • (ax + b) 3 a3 g ax + b 2(ax + b) 2 .
    [Show full text]
  • Antiderivatives Basic Integration Formulas Structural Type Formulas
    Antiderivatives Definition 1 (Antiderivative). If F 0(x) = f(x) we call F an antideriv- ative of f. Definition 2 (Indefinite Integral). If F is an antiderivative of f, then R f(x) dx = F (x) + c is called the (general) Indefinite Integral of f, where c is an arbitrary constant. The indefinite integral of a function represents every possible antideriv- ative, since it has been shown that if two functions have the same de- rivative on an interval then they differ by a constant on that interval. Terminology: When we write R f(x) dx, f(x) is referred to as the in- tegrand. Basic Integration Formulas As with differentiation, there are two types of formulas, formulas for the integrals of specific functions and structural type formulas. Each formula for the derivative of a specific function corresponds to a formula for the derivative of an elementary function. The following table lists integration formulas side by side with the corresponding differentiation formulas. Z xn+1 d xn dx = if n 6= −1 (xn) = nxn−1 n + 1 dx Z d sin x dx = − cos x + c (cos x) = − sin x dx Z d cos x dx = sin x + c (sin x) = cos x dx Z d sec2 x dx = tan x + c (tan x) = sec2 x dx Z d ex dx = ex + c (ex) = ex dx Z 1 d 1 dx = ln x + c (ln x) = x dx x Z d k dx = kx + c (kx) = k dx Structural Type Formulas We may integrate term-by-term: R kf(x) dx = k R f(x) dx 1 2 R f(x) ± g(x) dx = R f(x) dx ± R g(x) dx In plain language, the integral of a constant times a function equals the constant times the derivative of the function and the derivative of a sum or difference is equal to the sum or difference of the derivatives.
    [Show full text]
  • Lecture 15 Multiple Integration
    Lecture 15 Multiple Integration (Relevant section from Stewart, Section 15.1) We now turn to the integration of scalar-valued functions f : Rn R, i.e., f(x ,x , , c ), over → 1 2 · · · n regions in Rn. The need to perform such integrations is common in Physics. For example, we may wish to find: 1. The total charge Q in a region R R3 that is enclosed by a closed surface S, e.g., a spherical ⊂ surface of radius R centered at a point p R3, ∈ 2. The probability of finding an electron in the 1s state of a hydrogen atom within the distance α/a from the nucleus, where α 0 and a is the so-called Bohr radius. 0 ≥ 0 In all cases, the idea of integrating scalar-valued functions of several variables is a natural generalization of the integration of real-valued functions of a single variable, i.e., f : R R over the real line R. → We simply have to keep in mind the “Spirit of Calculus,” where we subdivide the region of interest and perform a summation over the subregions. Double Integrals (Integration in R2) (Relevant sections from Stewart, Section 15.1-2) We begin with the integration of functions of two variables, f(x,y), over regions in the plane R2. Suppose that a function f : R R2 is defined → over a region R R2. ⊂ Example: f(x,y) is the (2D or “areal”) density of a thin plate at (x,y), a measure of the amount of mass per unit area around the point (x,y).
    [Show full text]