21. Karl's Calculus Tutor - 11.3 Integration By Simple Substitution Section 11 Methods of integration. © 2001 by Karl Hahn, KCT logo.11.3 integration by Simple Subsitution. (For another view on http://www.karlscalculus.org/calc11_2.html

Section 11: Methods of Integration 11.3 Integration by Simple Subsitution (For another view on this topic see this tutorial by Stefan Waner and Steven R. Costenoble at Hoftra University When Dorothy visited Munchkin Land , it's a good thing for her that she arrived when she did. For on the day she arrived, the Inter-Oz Highway Act was still recent history. Had Dorothy arrived only a few months earlier, the Yellow Brick Road would still have been under construction, and Auntie Em would have had to await its completion before her niece could journey to The Emerald City and find her way back to Kansas. The Emerald City Public Works Commission had contracted the firm of Follup and Gollup Munchkin Construction to lay the section of the Yellow Brick Road that passed through Munchkin Land. The contract was very explicit. Not only did it require all the pavement to be of yellow brick, but the commission had, in its bureaucratic wisdom, also required that each course be precisely 25 bricks laid lengthwise across and that there be exactly 100 bricks per square meter of Yellow Brick Road. At first it seemed easy for Follup and Gollup to comply. They would just make the road 5 meters wide and use bricks that were 20 centimeters by 5 centimeters. So each meter of road would have 20 courses of 25 bricks each. Every 5 square meters of road had 500 bricks it met all the requirements. And without another thought they went and ordered bricks to those specifications.

22. ThinkQuest Library Of Entries calculus Limits. and Min. Values of Functions Point of Inflection Endpoint ExtremaDifferentials. Antiderivatives. Sigma Notation. Area under A Curve integration. http://library.thinkquest.org/10030/calcucon.htm

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Seeing is Believing , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Seeing is Believing click here Back to the Previous Page The Site you have Requested ... Seeing is Believing click here to view this site A ThinkQuest Internet Challenge 1997 Entry Click image for the Site Languages : Site Desciption Need a primer on math, science, technology, education, or art, or just looking for a new Internet search engine? This catch-all site covers them all. Maybe you're doing your homework and need to quickly look up a basic term? Here you'll find a brief yet concise reference source for all these topics. And if you're still not sure what's here, use the search feature to scan the entire site for your topic. Students Peter Oakhill College, Castle Hill Australia Suranthe H Oakhill College Australia Coaches Tina Oakhill College, Castle Hill

23. ThinkQuest Library Of Entries The fundamental theorem of calculus; Area and the definite integral; Trapezoidrule. Applications of integration Average value of a function; Area between two http://library.thinkquest.org/20991/calc/calc.html

Welcome to the ThinkQuest Internet Challenge of Entries The web site you have requested, Math for Morons like Us , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Math for Morons like Us click here Back to the Previous Page The Site you have Requested ... Math for Morons like Us click here to view this site A ThinkQuest Internet Challenge 1998 Entry Click image for the Site Languages : Site Desciption Have you ever been stuck on math? If it was a question on algebra, geometry, or calculus, you might want to check out this site. It's all here from pre-algebra to calculus. You'll find tutorials, sample problems, and quizzes. There's even a question submittal section, if you're still stuck. A formula database gives quick access and explanations to all those tricky formulas. Languages: English. Students J. Robert Davis High School Library UT, United States

24. UF/Statistics, Required And Elective Mathematics Courses Review of limits, differentiation, and integration; calculus of vector functions;line and surface integration; calculus of variations; Fourier series. http://www.stat.ufl.edu/academics/ugrad/undergrad_math.shtml

Required and Elective Mathematics Courses Credits: 4 Prereq: MAC 2311 or MAC 3472. Techniques of integration; applications of integration; differentiation and integration of inverse trigonometric, exponential and logarithmic functions; sequences and series. (M) Credits: 4 Prereq: MAC 2312 or MAC 3512 or MAC 3473. Solid analytic geometry; vectors; partial derivatives; multiple integrals. (M) MAS 2103 Matrices and Vector Spaces. (Fall, Spring) Credits: 3 Prereq: MAC 2311 or MAC 3472 or MAC 2233. Linear equations and matrices, elementary determinants, linear geometry of Euclidean spaces, vector spaces and linear transformations, eigenvalues. (M) MAS 3114 Computational Linear Algebra. (Fall, Spring, Summer) Credits: 3 Prereq: MAC 2312 (or MAC 3512 or MAC 3473) and a scientific programming language. Linear equations, matrices and determinants. Vector spaces and linear transformations. Inner products and eigenvalues. This course emphasizes computational aspects of linear algebra. (M) MAS 4105 Linear Algebra 1. (Fall, Spring, Summer) Credits: 4 Prereq: MAC 2313 or MAC 3474; MAS 3300 recommended.

25. Integration By Parts - HMC Calculus Tutorial integration by Parts We will use the Product Rule for derivatives to derive apowerful integration formula du = dx, v = e x. Then by integration by parts, http://www.math.hmc.edu/calculus/tutorials/int_by_parts/

Integration by Parts We will use the Product Rule for derivatives to derive a powerful integration formula: Start with [f(x)g(x)] = f(x)g (x) + f (x)g(x). Integrate both sides to get f(x)g(x) = f(x)g (x) dx + f (x)g(x) dx. Solve for f(x)g (x) dx, obtaining f(x)g (x) dx = f(x)g(x)- f (x)g(x) dx. This formula frequently allows us to compute a difficult integral by computing a much simpler integral. We often express the Integration by Parts formula as follows: Let u = f(x) dv = g (x) dx du = f (x) dx v = g(x) Then the formula becomes u dv u v v du To integrate by parts, strategically choose u, dv and then apply the formula. Example Let's evaluate xe x dx. Let u = x dv = e x dx du = dx v = e x Then by integration by parts, x e x dx x e x e x dx xe x -e x +C. A Faulty Choice A Reduction Formula Integration by parts ``works'' on definite integrals as well: b a u dv = uv b a b a v du. Example We will evaluate arctan(x) dx. Let u = arctan(x) dv = dx du = 1+x dx v = x Then by integration by parts, arctan(x) dx x arctan(x) x 1+x dx x arctan(x) ln (1+x p ln(2)-0 p - ln( Sometimes it is necessary to integrate twice by parts in order to compute an integral: Example Let's compute e x cos(x) dx.

26. Mulitple Integration - HMC Calculus Tutorial Multiple integration Recall dy. where the limits of integration aredetermined by the region R over which we are integrating. Notes. http://www.math.hmc.edu/calculus/tutorials/multipleintegration/

Multiple Integration Recall our definition of the definite integral of a function of a single variable: Let f(x) be defined on [a,b] and let x ,x ,x n be a partition of [a,b]. For each [x i-1 ,x i ], let x i [x i-1 ,x i ]. Then b a f(x) dx = lim max D x i n i = 1 f(x i D x i Take a quick look at the Riemann Sum Tutorial We can extend this definition to define the integral of a function of two or more variables. Double Integral of a Function of Two Variables Let f(x,y) be defined on a closed and bounded region R of the xy-plane. Set up a grid of vertical and horizontal lines in the xy-plane to form an inner partition of R into n rectangular subregions R k of area D A k , each of which lies entirely in R. (Ignore the rectangles that are not entirely contained in R) Choose a point (x k , y k ) in each subregion R k . The sum n k = 1 f(x k , y k D A k is called a Riemann Sum . In the limit as we make our grid more and more dense, we define the double integral of f(x,y) over R as R f(x,y )dA = lim max D A k n k = 1 f(x k , y k D A k Notes If this limit exists, we say that f is integrable over the region of integration R.

27. Calculus calculus. X = gradient (M) calculates the one dimensional gradient if M isa vector. integration. count, extra/integration/count.m. no description http://octave.sourceforge.net/index/calculus.html

Calculus Differentiation Integration Ordinary differential equations Differentiation [main/general/del2.m] D = del2 (M) calculates the Laplace Operator diff If X is a vector of length N, `diff (X)' is the vector of first differences X(2) - X(1), ..., X(n) - X(n-1). gradient [main/general/gradient.m] X = gradient (M) calculates the one dimensional gradient if M is a vector. Integration count [extra/integration/count.m] no description cquadnd [extra/integration/cquadnd.m] number of dimensions to integrate crule [extra/integration/crule.m] This function computes Gauss-Chebyshev base points and weight factors [extra/integration/crule2d.m] no description [extra/integration/crule2dgen.m] no description dblquad not implemented gquad [extra/integration/gquad.m] This function evaluates the integral of an externally defined function fun(x) between limits xlow and xhigh. [extra/integration/gquad2d.m] This function evaluates the integral of an externally [extra/integration/gquad2d6.m] ==== Six Point by Six Point Double Integral Gauss Formula ==== [extra/integration/gquad2dgen.m]

28. Bigchalk: HomeworkCentral: Integration (Pre-Calculus & Calculus) HomeworkCentral Linking Policy. HIGH SCHOOL BEYOND Mathematics Precalculus calculus integration. INTRODUCTION General; Area; http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/Homework/High_School/Mat

Home About Us Newsletters Log In/Log Out ... Product Info Center Email this page to a friend! K-5 Integration document.write(''); document.write(''); document.write(''); document.write(''); INTRODUCTION General Area Fundamental Theorem of Calculus ... Contact Us

29. World Web Math: Calculus Summary An overview of calculus ideas. Covered are derivative rules and formulas as well as some basic integration rules. http://web.mit.edu/wwmath/calculus/summary.html

Calculus Summary Calculus has two main parts: differential calculus and integral calculus. Differential calculus studies the derivative and integral calculus studies (surprise!) the integral. The derivative and integral are linked in that they are both defined via the concept of the limit: they are inverse operations of each other (a fact sometimes known as the fundamental theorem of calculus): and they are both fundamental to much of modern science as we know it. Derivatives The limit of a function f x ) as x approaches a is equal to b if for every desired closeness to b , you can find a small interval around (but not including) a that acheives that closeness when mapped by f . Limits give us a firm mathematical basis on which to examine both the infinite and the infinitesmial. They are also easy to handle algebraically: where in the last equation, c is a constant and in the first two equations, if both limits of f and g exist. One important fact to keep in mind is that doesn't depend at all on f a ) in fact

30. Bigchalk: HomeworkCentral: Pre-Calculus & Calculus (Mathematics) integration by Parts Drill; Multivariable calculus Practice; Sequences Series;TI82 Graphing Calculator Tutorial; Word Problem Practice (Grades 5 - 12). http://www.bigchalk.com/cgi-bin/WebObjects/WOPortal.woa/wa/BCPageDA/sec~CAB~9353

Home About Us Newsletters Log In/Log Out ... Product Info Center Email this page to a friend! K-5 document.write(''); document.write(''); document.write(''); document.write(''); PRE-CALCULUS: INTRODUCTION TO CALCULUS General Trigonometry Tutorial/Practice ... Contact Us

31. STSC CrossTalk - Measuring Calculus Integration Formulas Using Function Point An Measuring calculus integration Formulas Using Function Point Analysis Nancy Redgate,American Express Risk Management Dr. Charles Tichenor, Defense Security http://www.stsc.hill.af.mil/crosstalk/2002/06/redgatetichenor.html

var root = "/"; var menuRoot = "/menus"; var imageRoot = "/images"; Entire Site CrossTalk Only document.searchForm.CiScope.selectedIndex=1; Mission Staff Contact Us Subscribe Now ... CrossTalk Jun 2002 Jun 2002 Issue Measuring Calculus Integration Formulas Using Function Point Analysis Nancy Redgate, American Express Risk Management Dr. Charles Tichenor, Defense Security Cooperation Agency Function point counters, software developers, and others occasionally need to measure the size and complexity of calculus integration formulas embedded in engineering and scientific applications. Sizing these formulas using function point analysis can result in more accurate measures of application size and improved quality in forecasting costs, schedule, and quality. It can also improve the confidence of those new to the function point methodology as they see that all of their calculus work is recognized and measured. This article shows an approach to sizing these formulas. This methodology is in full compliance with the International Function Point Users Group procedures and does not require any additional counting rules or patches. Measuring the size and complexity of software is critical to the development of business models that accurately forecast the development cost, duration, and quality of future software applications. One way to measure software size is through function point analysis, especially using the methodology of the International Function Point Users Group (IFPUG) as published in their "Function Point Counting Practices Manual" (CPM) version 4.1 [1]. Readers unfamiliar with function point analysis are referred to the CPM as the primary reference for this methodology.

32. Knowledge Bank Calculus List A more difficult example of integration by Parts, Joanne Spiers,2 Aug 99. How to do integration by Parts, Mike Busfield, 27 July99. http://www.maths-help.co.uk/Knowldge/Calc/List.htm

Knowledge Bank Contents List for Questions and full solutions are posted on the following topics: Question Sent by Date How to differentiate x by first principles Martin Bland 28 Aug 99 A more difficult example of Integration by Parts Joanne Spiers 2 Aug 99 How to do Integration by Parts Mike Busfield 27 July 99 Area of a circle by integration Cristal 30 Jun 99 Integrating squared trig functions Jen Wilcox 9 May 99 Find the minimum surface area of a tank with given volume Arron Charman 15 Apr 99 Maximum area of an isosceles triangle with a fixed perimeter J W 12 Mar 99 Why is my answer for the area by integration wrong? Jon Grundy 27 Feb 99 Differentiating sin( x ) by first principles J Sanders 17 Feb 99 Minimum surface area of a cylinder with a given volume Natasha Squires 14 Feb 99

33. Mathematics/Course Descriptions: 2002 - 2003 Undergraduate Catalog Review of limits, differentiation and integration; calculus of vector functions;line and surface integration; calculus of variations; Fourier series. http://www.reg.ufl.edu/02-03-catalog/courses/m_o/courses_Mathematics.htm

Course Descriptions Reading a Course Description Entry Index to Course Prefixes Florida's Statewide Numbering System Staff and Faculty ... Catalog Home Course Descriptions A's B D E's F G H L M O P R S Z Course Descriptions Mathematics College of Liberal Arts and Sciences INSTRUCTIONAL STAFF Alladi, K., Chair; Block, L.S., Associate Chair; Berkovich, A.; Bona, M.; Boyland, P.L.; Brechner, B.L.; Brooks, J.K.; Carter, C.; Cenzer, D.S.; Chen, Y.; Crew, R.M.; Dinculeanu, N.; Drake, D.A.; Dranishnikov, A.N.; Edwards, B.H.; Ehrlich, P.E.; Emch, G.G.; Garvan, F.G.; Glover, J.; Gopalakrishnan, J.; Groisser, D.J.; Hager, W.W.; Ho, C.Y.; Hueter, I.; Keating, K.P.; Keesling, J.E.; Khuri, R.L.; King, J.L.; Klauder, J.R.; Kutuzova, M.; Larson, J.A.; Levin, N.; Mair, B.A.; Martinez, J.; McCracken, D.L.; McCullough, S.A.; Metzler, D.; Mitchell, W.J.; Moore, T.O.; Moskow, S.; Olson, T.E.; Pilyugin, S., Pop-Stojanovic, Z.R.; Rao, M.K.; Robinson, P.L.; Rudyak, Y.; Saxon, S.A.; Shen, L.C.; Sin, P.K.; Smith, J.Y.; Smith, R.; Summers, S.J.; Thompson, J.G.; Tiep, P.H.; Tornwall, S.B.; Townsend, M.D.; Turull, A.; Vince, A.J.; Voelklein, H.K.; Walsh, T.; White, N.L.; Wilson, D.C.; Zapletal, J. Undergraduate Coordinator: TBA Actuarial Science Advisers: Bruce Edwards, 364 Little Hall, 392-0281 ext 281;

34. PinkMonkey.com Calculus Study Guide - Section 6.3 Methods Of Integration 6.3 Methods Of integration. There are special methods of reducing an integralto a standard form. The following are the methods of integration. http://www.pinkmonkey.com/studyguides/subjects/calc/chap6/c0606301.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey There are special methods of reducing an integral to a standard form. The following are the methods of integration. next page Index 6.1 Anti-derivatives (indefinite Integral) 6.2 Integration Of Some Trignometric Functions 6.3 Methods Of Integration 6.4 Substitution And Change Of Variables 6.5 Some Standard Substitutions 6.6 Integration By Parts 6.7 Integration By Partial Fractions ... Chapter 7 4652 PinkMonkey users are on the site and studying right now.

35. PinkMonkey.com Calculus Study Guide - Section 6.6 Integration By Parts 6.6 integration by Parts. This method is used to integrate a product,although one of the factors of the product could be unity. http://www.pinkmonkey.com/studyguides/subjects/calc/chap6/c0606601.asp

Support the Monkey! Tell All your Friends and Teachers Get Cash for Giving Your Opinions! Free College Cash 4U Advertise Here See What's New on the Message Boards today! Favorite Link of the Week. Your house from the sky! ... Tell a Friend Win a $1000 or more Scholarship to college! Place your Banner Ads or Text Links on PinkMonkey for $0.50 CPM or less! Pay by credit card. Same day setup. Please Take our User Survey This method is used to integrate a product, although one of the factors of the product could be unity. Also note that one of the product of functions can be easily integrable. The factors of the product will be two different kinds of functions. They will be : Theorem : Here u is called the first function which is to be differentiated and v is the second function which is to be integrated. Rule for the proper choice of the first function : Let us denote the various types of functions that we come across by the first alphabet i.e. L. I. A. T. E. for logarithmic, Inverse circular, Algebraic, Trigonometric (circular) and Exponential respectively. The first function to be selected will be the one which comes first in the order of the word " LIATE " next page Index 6.1 Anti-derivatives (indefinite Integral)

36. Calculus - Integration By Parts - Technical Tutoring cover Schaum's Easy Outline calculus A simplified and updated version of the classic Keywordsintegral, integration, technique, parts, products, tabular, uv. http://www.hyper-ad.com/tutoring/math/int_parts.htm

Formula Tabular Method u-v Method Examples ... Gift Shop Integration By Parts Suppose we have two functions multiplied by each other and differentiate according to the product rule: then by integrating both sides between the limits a and b and rearranging gives or as the formula is better known INTEGRATION BY PARTS back to top Tricks: If one of the functions is a polynomial (say nth order) and the other is integrable n times, then you can use the fast and easy Tabular Method: Tabular Method Suppose and . Then if we set up a table, differentiating f(x) as many times as it takes to get to zero and integrating g(x) as many times, we get D I (a) (b) (a) (c) (b) (c) This method is much faster than the f-g method or the older u-v, especially for iterated (more than once) integrals by parts (Thanks to Dr. William T. Guy, UT Austin). back to top Advanced There is a way to extend the tabular method to handle arbitrarily large integrals by parts - you just include the integral of the product of the functions in the last row and pop in an extra sign (whatever is next in the alternating series), so that The trick is to know when to stop for the integral you are trying to do. Try it for a few simple functions, you'll see!

37. Calculus Made Easier: A Calculus Tutorial cx n , then f(x) dx = cx n+1 /(n+1). Some other integration formulas If Library Grolier'sHistory of Math SOS Math Homework Help Dan's Math calculus Books The http://www.wtv-zone.com/Angelaruth49/Calculus2.html

Go to Links Page 2 Integration Persimmons Tree Integration is the measure of the area under a curve. What it does is to take the sum of rectangles under the curve. As more rectangles are inscribed, but with smaller area, we get a better approximation. I don't have a picture of this particular item to show you, but I have something that will illustrate my point. You can see that for more triangles that are in that circle, the closer to the area of the circle the inscribed polygon is. This is also true for inscribed rectangles under any curve drawn in the cartesian coordinates. So, the integration is the sum of rectangles inscribed under any graph, where each rectangle is f(x n )(x n -x n-1 ) for any point along the curve. As I said earlier, integration is an inverse function of derivatives, so they're somewhat related. The symbol for integration is a flattened capital S, which stands for summa in Greek. The integration of f(x) is written as f(x) dx, where dx is the difference between x n and x n-1 . See, the examples below. If f(x) = x n , then f(x) dx = x n+1 /(n+1).

38. MATH118 - Introductory Calculus, Part II: Integration And Its Applications Year 2002/2003 Introductory calculus, Part II integration and ItsApplications MATH 118 SP. This course continues MATH117. It is http://www.wesleyan.edu/course/math118s.htm

document.domain="wesleyan.edu"; Wesleyan Home Page WesMaps Home Page WesMaps Archive Course Search ... Course Search by CID Academic Year 2002/2003 Introductory Calculus, Part II: Integration and Its Applications MATH 118 SP This course continues MATH117. It is designed to introduce basic ideas and techniques of calculus. Students should enter MATH118 with sound precalculus skills and with very limited or no prior study of integral calculus. Topics to be considered include differential and integral calculus of algebraic, exponential, and logarithmic functions. MAJOR READINGS The Harvard Consortium Text CALCULUS by Hughes-Hallett, Gleason, et al. EXAMINATIONS AND ASSIGNMENTS Regular assignments, occasional projects and tests, and a final examination. ADDITIONAL REQUIREMENTS and/or COMMENTS A graphing calculator will be required. Classes will be a combination of lecture/presentation and work in small groups or labs. The format is non-traditional, with emphasis on group work and frequent use of technology. Regular attendance and participation is essential. Students must have taken MATH 117. Unless preregistered students attend the first class meeting or communicate directly with the instructor prior to the first class, they will be dropped from the class list. NOTE: Students must still submit a completed Drop/Add form to the Registrar's Office.

39. MATH118 - Introductory Calculus, Part II: Integration And Its Applications; Infi CID Introductory calculus, Part II integration and its Applications;Infinite Series MATH118 SP. This course continues MATH117. http://www.wesleyan.edu/wesmaps/course9900/math118s.htm

document.domain="wesleyan.edu"; Wesleyan Home Page WesMaps Home Page Course Search Course Search by CID Introductory Calculus, Part II: Integration and its Applications; Infinite Series MATH118 SP This course continues MATH117. It is designed to introduce basic ideas and techniques of calculus. Students should enter Math118 with sound precalculus skills and with very limited or no prior study of integral calculus. Topics to be considered include differential and integral calculus of algebraic, exponential, and logarithmic functions. MAJOR READINGS The Harvard Consortium Text CALCULUS by Hughes-Hallett, Gleason, et al. EXAMINATIONS AND ASSIGNMENTS Regular assignments, occasional projects and tests, and a final examination. ADDITIONAL REQUIREMENTS and/or COMMENTS A graphing calculator will be required. Classes will be a combination of lecture/presentation and work in small groups or labs. The format is non-traditional, with emphasis on group work and frequent use of technology. Regular attendance and participation is essential. Unless preregistered students attend the first class meeting or communicate directly with the instructor prior to the first class, they will be dropped from the class list. NOTE: Students must still submit a completed Drop/Add form to the Registrar's Office.

40. Calculus/Integration calculus/integration. 1. The process of indefinite integration finds a function(F) whose derivative is a given function (f), call it f(V). If. then. http://www.sp.uconn.edu/~cdavid/mathrev2/node7.html

Next: Multidimensional Integrals Up: Math Review Previous: Calculus/Partial Differentiation Calculus/Integration The process of indefinite integration finds a function (F) whose derivative is a given function (f), call it f(V). If then where C is a constant of integration. If then (since ) where, again, C is a constant of integration (which vanishes upon differentiation of p(V) to re-obtain f(V). The process of definite integration means, given a function f(V) and two values of V, finding the area under the graph of f(V) between the aforementioned values of V: where, of course, the constant of integration cancels. a special integral is defined: There exist a special set of indefinite and definite integrals which should be known: (a) (b) (c) The process of indefinite integration finds a function (F) whose derivative is a given function (f), call it f(V). If then where C is a constant of integration. If then (since ) where, again, C is a constant of integration (which vanishes upon differentiation of p(V) to re-obtain f(V). The process of definite integration means, given a function f(V) and two values of V, finding the area under the graph of f(V) between the aforementioned values of V: where, of course, the constant of integration cancels. a special integral is defined:

Page 2     21-40 of 86    Back | 1  | 2  | 3  | 4  | 5  | Next 20