#### Calc I - Final Paper [1]

Rowan, MATH 1
Excerpt: ... the slope (rise/run), Leibniz created an equation that states ds = dy/dx. So as mentioned earlier, Leibniz showed that the tangent line at a point can be found from dy/dx. The next step for Leibniz was finding an easier way to calculate area under a curve. During the time of Leibniz, mathematicians were still using the process of Archimedes, which involved finding the areas of geometric shapes that could be inscribed under the curve (rectangles became the shape of choice as opposed to triangles that Archimedes used). Riemann sum s are the most popular version of this and is a method for approximating the values of integrals. To calculate Riemann sum s, one must first have a curve and the two endpoints of the curve. These two endpoints can be denoted by a and b. Next, the graph must be divided into n equally divided intervals (starting with x0, each successive point will be x1, x2 xn). By dividing into equally divided intervals, a partition is created. If P represents the partition with n subin ...

#### wk6calclect1

Allan Hancock College, MAT 1102
Excerpt: ... MAT1102(64612) Algebra&Calculus 1, S1 2004 Week 6 First Lecture in Calc1 ' \$ Calculus Week 6 - Lecture 1 - Outline Matlab Notes Revision - Riemann Sum s Slide 1 Revision - Definite Integral as Area - Example Study Book 3.3 Interpretations of Definite Integral Notation Units Average of a Function Applications of Definite Integrals & % \$ ' Matlab Notes Left Riemann Sum for Slide 2 2 0 e-x sin x dx n=1001; % number of points, ie n-1 subintervals a=0;b=2*pi; h=(b-a)/(n-1); % step size x=linspace(a,b-h,n); % from x=a to x=b-h for left sum y=exp(-x).*sin(x); leftsum=sum(y*h) & % MAT1102(64612) Algebra&Calculus 1, S1 2004 Week 6 First Lecture in Calc2 ' \$ Revision - Riemann Sum s n-1 Slide 3 Left sum = i=0 n f (ti )t Right sum = i=1 f (ti )t ' & % \$ Revision - Definite Integral as Area Example Use a plot to decide if Slide 4 Above axis - positive Below axis - negative Conclusion: 2 0 2 0 e-x sin x dx is positive or negative Look at plot of y = e-x sin x e-x sin x dx is _ ...

#### hmwk4

Virginia Tech, MATH 1206
Excerpt: ... Homework 4 1.Forthefollowingtableofvaluesfind: a.theleftRiemannsumSl b.therightRiemannsumSr c.theRiemannsumSmaxofmaximumvalues d.theminimumRiemannsumSminofminimumvalues 2.FindthefollowingRiemannsumsfor g(x) = 1 5x + 1 2 overtheinterval[1,2]di ...

#### Guide4

University of Illinois, Urbana Champaign, MATH 220
Excerpt: ... ummation. Some basic sums you need to know in Theorem 2.1. Basic properties of sums. The Principle of Mathematical Induction. Section 4.3 Area. Using left-hand and right-hand sums to approximate area under a curve. More rectangles give better approximations. The definition of area under the curve and its relation to Riemann Sum s. Why we can pick any point in a subinterval in a Riemann Sum . 1 Section 4.4 The Definite Integral. How sums, area under the curve and the integral relate to each other. The definition of the definite integral, how this relates to Riemann Sum s. The idea of signed or algebraic area versus total area, why this idea is useful. How to compute definite integrals from the definition. Basic properties of the definite integral. Average values, the formula and what it means physically. The Integral Mean Value Theorem. Section 4.5 The Fundamental Theorem of Calculus. How antiderivatives and the area under the curve relate. How antiderivatives and derivatives relate to each other, effectivel ...

#### TI83-Riemann

Arizona, M 129
Excerpt: ... 7.5 VARIOUS RIEMANN SUM S (implemented for TI83) In order to understand Riemann sum s, which are sums, one has to understand how a particular programming language does the summation. So the first example shows one possible way of ADDING the first N whole numbers, that is, computes the sum 1+2+3+.+N. Pay attention to the commands. A. Program to add the sum of first N whole numbers PROGRAM SUM :Prompt N :0->S :For(I,1,N) :S+I->S :End :Disp "SUM UP TO N" :Disp S B. Program computing the Left, Mid, Right, Trapezoid, and Simpson Riemann sum s (do not introduce what is after the %sign; I wrote those notes as a reminder of what the variables represent; before running the program RIEMANN introduce the function that you want to integrate as Y1 in the "Y=" menu) PROGRAM RIEMANN :Prompt A,B,N :(B-A)/N->H %(H is the length of each subinterval) :0->L %(L will give the left Riemann sum ) :0->R %(R will give the right Riemann sum ) :0->M %(M will give the mid Ri ...

#### M220-24

University of Illinois, Urbana Champaign, MATH 220
Excerpt: ... Math 220, Loeb Lecture 24, October 18, 2006 1 1. Come to Class We now cover material in a way that is not in the book. Come to class. 2. A Tutoring Room is Open 7 p.m, Monday, Tuesday, Wednesday, Thursday, Room 140 Lincoln Hall. 9 3. Homework 16 due Thursday, October 19 at 9 a.m. Section 4.5: #8, 10, 18, 28. Section 4.6: #24, 30, 32, 78, 80, 82. 4. Homework 17 due Tuesday, October 24 at 9 a.m. Section 4.7: #2, 8, 34, 36. Section 4.8: #8, 22, 24, 36. Section 4.9: #6, 10. 5. Written problem for next week Find the upper sum, the lower sum, the Riemann sum (evaluating at the left of each interval) and the value of Ef ( x) for the function f (x) = x4 on the interval [ 1; 1] using x = 3=10. Remember, this is a sample problem for the next exam. 6. Area under graphs. Another name for the antiderivative of a function is the inde.nite integral. We will see the reason for this later. Now we study the de.nite integral or just integral. The idea of a de.nite integral is most easily illustrated by the problem of ...

#### Lab12RiemannSums

MN State, M 260
Excerpt: ... Name:_ Math 260 Riemann Sum s Activity Lab 12 For questions 1, 2, and 3(a), use the Riemann sum s tutor, found on the main toolbar of Maple at Tools - Tutors - Calculus-Single Variable - Riemann Sum s (or Tools - Tutors - Calculus-Single Variable - Approximate Integration). 1. Approximate the area under the graph of f ( x) x 4 2cos(3x), x 2.5, 4 . Display the result five times using random and n = 10. Record the results below. (Note that this is not actually a fully random Riemann Sum since the partition size does not vary.) (i) (ii) (iii) (iv) (v) Are the approximations reasonable? 2. Find a Riemann sum for the function f ( x) x x 2 4 on each of the listed intervals. Display the result using random and n = 10. Do the sum three times on each interval. Record the results below. (a) [-2, 2] i) ii) iii) (b) [-3, 2] i) ii) iii) (c) [-2, 3] i) ii) iii) Explain why some values are negative, positive, or (approximately) zero. 3. Consider the area und ...

#### hw8

Wisconsin, MATH 221
Excerpt: ... Math 221 Differential and Integral Calculus I Lecture 2, Fall 2007 Homework assignment 8. 1. Use Riemann sum s with four intervals of length one to find positive numbers L and U with 5 1 3+ dx U. 3<L x 1 2. Let f (x) = 3x + 4, a = 2, b = 5. The region a x b, 0 y f (x) is a trapezoid. b (i) Draw the trapezoid and find the area a f (x) dx of this trapezoid by elementary geometry. (ii) Evaluate the approximation 6 f (cj )xj and the mesh P when j=1 x0 = 2, c1 = 2.2, x1 = 2.5, c2 = 2.8, x2 = 3.0 c3 = 3.2, x3 = 3.5, c4 = 3.9, x4 = 4.1 c5 = 4.2, x5 = 4.4, c6 = 4.8, x6 = 5.0. Draw a graph illustrating the trapezoid and the six rectangles xj-1 x xj , 0 y f (cj ) whose areas f (cj ) xj sum to the approximation just computed. (iii) Evaluate the approximation 8 f (cj )xj when j=1 k xj = a + (b - a), 8 If you like, you may use the formula n j=1 j cj = = n(n+1) . 2 xj-1 + xj . 2 2. Let 0 < a < b. Compute the integral b x3 dx a using explicit approximation by Riemann sum s. (You may consult Problem 3 on the th ...

#### wk5calclect2

Allan Hancock College, MAT 1102
Excerpt: ... MAT1102 Algebra&Calculus 1, S1 2004, Week 5 Second Lecture in Calc 1 ' \$ Calculus Week 5 - Lecture 2 - Outline Matlab Notes - differencing for the derivative The Precise Distance Travelled - The Bug problem Study Book 3.2 The Definite Integral Slide 1 Left sum Right sum Limits of Riemann sum s - the Definite Integral Computing Definite Integrals Definite Integrals as area Example Readings and Exercises ' & % \$ Matlab Notes Approximating Derivatives by Differencing % plot of the derivative of y = 1/(1+t) from 0 to 1 h=0.01; % the step size t1=0:h:1; % the domain of 101 points t2=t1+h; % a domain of 101 points shifted slightly to the right y1=1./(1+t1); % 101 function values at t1 y2=1./(1+t2); % 101 function values at t1+h deriv=(y2-y1)/h; % approx to the deriv at 101 points plot(t1,deriv); % Note: t1 or t2 could be used here & % Slide 2 MAT1102 Algebra&Calculus 1, S1 2004, Week 5 Second Lecture in Calc 2 ' \$ The Precise Distance Travelled - a Bug Problem Problem: bug crawls at speed ...

#### wk12calc18

Allan Hancock College, MAT 1102
Excerpt: ... Approximate Integration Some functions cannot be integrated analytically. Approximate Integration Lecture 18 Approximate integration is what we have been doing with Matlab using the sum() function. We will only consider three main methods of approximate integration, with a brief mention of the Trapezoidal Rule. Left Endpoint Riemann sum s Right Endpoint Riemann sum s Midpoint Riemann sum s Algebra & Calculus I Example 2 Find estimates for 0 e-x dx and compare with the answer 0.8821 2 (correct to 4 decimal places). Note: that this is one of those integrals which we cannot find by standard means. Substitution will not work. First plot the curve. Calculus: Approximate Integration Lect. 18 1 / 16 Calculus: Approximate Integration Lect. 18 4 / 16 Approximation Methods y 1 The Left Endpoint Riemann Sum in Matlab Left Endpoint Approximation Left Sums 0 2 0 Right Sums 1 e-x dx 2 2 x h=0.1; x=0:h:2-h; y=exp(-x.^2); left=sum(y*h) % % % % your choice of step length the left hand endpoints the values at th ...

Hobart and William Smith Colleges, MATH 131
Excerpt: ... Math 131 Calculus II: Riemann Sum s To compute a Riemann sum , select " Riemann Sum s Utility" from the menu at the top. Additional instructions below. Riemann Sum s Utility xmin -1 xmax 1 ymin -0.2 ymax 1.2 Intervals 512 Left Endpoints: 1.57065279 Right Endpoints: 1.57065279 Midpoints: 1.57083836 ~Circumscribed: 1.57455904 ~Inscribed: 1.56674654 Trapezoids: 1.57065279 (Try clicking on the graph!) Enter a function of x: y = sqrt(1-x^2) Compute! Clear Intervals Divide Intervals Select Display: Left Endpoints Quick Instructions General: When the applet starts up, it is showing a "Main Screen" where you can see the graph of functions. The available functions are shown in a list on the left. Click on a function to view its graph. You can define your own functions to add to this list using the three buttons at the lower left of the Main Screen. There are three ways to define new functions: using expressions (such as 0.5*x^2+sin(3*x-2), by giving the graph of the function, or by listing a table of (x,y) pairs. T ...

#### q8

Los Angeles Southwest College, M 122
Excerpt: ... Math 122 (Section 11) Name Quiz 8 November 13, 2007 1. Due to a terrible storm, the water in a river is rising. The people in a nearby town are worried that flooding will occur. They have sandbags along both sides of the river which will stop the flooding as long as the total rise in the water-level is less than 15 inches. The rate at which the water is rising is decreasing until the storm finally stops 24 hours later, and is recorded every 6 hours in the table below. # hours # inches per hour 0 6 12 18 24 1.5 0.6 0.3 0.2 0 (a) (5 points) By computing a left Riemann sum , a right Riemann sum , and then averaging these values, give three different estimates for the number of inches that the water-level of the river rises during this 24-hour period. (b) (1 point) Circle the statement which is most accurate for this 24-hour period. Note: You will not get credit for this problem unless your estimates in part (a) are correct. i. The town definitely floods. ii. The town probably floods, but there is a slight ch ...

#### hmwk4ans

Virginia Tech, MATH 1206
Excerpt: ... Homework 4 Answers 1.Forthefollowingtableofvaluesfind: a.theleftRiemannsumSl b.therightRiemannsumSr c.theRiemannsumSmaxofmaximumvalues d.theminimumRiemannsumSminofminimumvalues a.7.8 b.6.8 c.6.6 d.8 2.FindthefollowingRiemannsumsfor g(x) ...

#### lec36

Vanderbilt, MATH 155b
Excerpt: ... nite. The importance of convergence is illustrated here by the example of the geometric series. If a = 1, S = 1 + 1 + 1 + . = . But S aS = 1 or = 1 does not make sense and is not usable! Another type of series: 1 np n=1 We can use integrals to decide if this type of series converges. First, turn the sum into an integral: 1 np n=1 1 dx xp If that improper integral evaluates to a nite number, the series converges. Note: This approach only tells us whether or not a series converges. It does not tell us what number the series converges to. That is a much harder problem. For example, it takes a lot of work to determine 1 2 = n2 6 n=1 Mathematicians have only recently been able to determine that 1 n3 n=1 converges to an irrational number! Harmonic Series 1 n n=1 1 dx x We can evaluate the improper integral via Riemann sum s. Well use the upper Riemann sum (see Figure 1) to get an upper bound on the value of the integral. 2 Lecture 36 18.01 Fa ...

#### Lesson 2

University of Illinois, Urbana Champaign, EDUC 362
Excerpt: ... Defining and Computing Definite Integrals Overview: In this lesson, students will learn what a Riemann sum is and be given a stepby-step procedure of how to formulate them. They will also learn how to calculate both upper and lower Riemann sum s. They will examine Riemann sum s in depth through geometer's sketchpad. A formal definition of a definite integral will be discussed and students will learn and use integral notation integrand, limits of integration, variable of integration, and what it means for an integral to be Riemann integrable. The students will also discover the properties of definite integrals and explore how to use graphs to calculate definite integrals over intervals. Finally, they will learn how to calculate the worst possible error of an approximation integral. Grade Level/Subject: This lesson is for 12th graders in AP Calculus. Time: 3-50 minute class periods Purpose: The purpose of this lesson is to define definite integrals using Riemann sum s. By doing this, students will truly understa ...

#### MAT1102_calc_l20_wk13_2solutions

Allan Hancock College, MAT 1102
Excerpt: ... 13 part 2) Slide 6 Integration Using Tables Find Number of the list of standard integrals (back of textbook) is: Complete the square on x2 +4x+5 to make it look like a2 +u2. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 7 Integration Using Tables Then substitute if necessary to simplify. Then use the standard integral. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 8 Approximate Integration Left Endpoint Riemann sum s h=0.1; x=0:h:2-h; y=exp(-x.^2); left=sum(y*h) Right Endpoint Riemann sum s h=0.1; x=h:h:2; y=exp(-x.^2); right=sum(y*h) Midpoint Riemann sum s h=0.1; x=h/2:h:2-h/2; y=exp(-x.^2); mid=sum(y*h) Trapezoidal rule average of left and right sums, trap = (left+right)/2 MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 9 Example Find estimates for the value of Note that this is one of those integrals we cannot find by standard means. Substitution won't work. First plot the curve. The value correct to ...

#### MAT1102_calc_l20_wk13_2

Allan Hancock College, MAT 1102
Excerpt: ... Lecture 20 (week 13 part 2) Slide 6 Integration Using Tables Find Number of the list of standard integrals (back of textbook) is: Complete the square on x2 +4x+5 to make it look like a2 +u2. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 7 Integration Using Tables Then substitute if necessary to simplify. Then use the standard integral. MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 8 Approximate Integration Left Endpoint Riemann sum s h=0.1; x=0:h:2-h; y=exp(-x.^2); left=sum(y*h) Right Endpoint Riemann sum s h=0.1; x=h:h:2; y=exp(-x.^2); right=sum(y*h) Midpoint Riemann sum s h=0.1; x=h/2:h:2-h/2; y=exp(-x.^2); mid=sum(y*h) Trapezoidal rule average of left and right sums, trap = (left+right)/2 MAT1102 Algebra & Calculus I Calculus Lecture 20 (week 13 part 2) Slide 9 Example Find estimates for the value of Note that this is one of those integrals we cannot find by standard means. Substitution wont work. First plo ...

#### takehome3

Wilfrid Laurier, MATH 251
Excerpt: ... Take Home Quiz 2: Riemann Sum s Due Wednesday, December 8 Math 251 Lecture 01 [5 pts] You will have only one chance to complete this quiz. Good luck! 1. Evaluate the sum n i2 + 2i + 1 i=0 [5 pts] in terms of n. (Hint: dont forget that your sum formulas apply to sums which begin at i = 1.) 2. Express the limit below as a denite integral, but do not evaluate it. n n lim i=1 sin n i n [5 pts] 3. Find the area bounded by x = 2, x = 5, y = 2x2 + 2 and the x axis by computing an appropriate Riemann sum (you may NOT use the Fundamental Theorem of Calculus as a short cut). 1 ...

#### q11

Penn State, MATH 140
Excerpt: ... Math 140, Section 5 Quiz #11 April 6, 2001 Note: Calculators are not allowed. 1. Let f (x) = x + 1. Consider the interval [a, b] where a = 1, b = 2. Form a Riemann sum with three intervals of equal length, with x1 , x2 , x3 being the right endpoints of the intervals. Evaluate the sum. 2 2. Evaluate the definite integral 1 x + 1 dx. ...

#### ws12

Maryland, MATH 12
Excerpt: ... MATH 141, Worksheet 12: 9.2, 9.3, and 9.4, Sequences and Series (1) Consider the following recursively dened sequence, a0 = 2, an+1 = sin(an ), for all n 1. (a) It can be shown that this sequence is bounded and decreasing. Explain why it converges to a limit L (you must site the appropriate theorem). (b) Conjecture the value of L. Justify your answer. (Hint: Look at pg 589 in your textbook). (2) (a) Give an example of a series for which limn an = 0 , but (b) Determine if the following series diverges. an diverges. tan1 n n=0 (3) The following exercise relates certain Riemann Sum s with special sequences. (a) Let 2 3 n 1 + 2 + 2 + + 2 for n 1. 2 n n n n 1 Show that an is a Riemann sum for 0 x dx for each n 1. Then nd limn an . (Hint: Divide the interval [0, 1] into n 1 subintervals, and use the right endpoint rule. Essentially x = 1/n. Remember the Riemann Sum is equal to the integral for when x 0. Note this happens when n ). an = (b) Let bn ...

#### RiemannSumNotes

UT Arlington, M 5337
Excerpt: ... Note that I did not get to the terminology " Riemann sum " in class. However, the sums L ( n ) and R ( n ) are Riemann sum s for f over the interval [ a, b ] as shown in class. Notice that later when we computed LP ( n ) and RP ( n ) that these are also Riemann sum s for f over the interval [ a, b ] . There are many such sums, depending on the partition P we choose and the choices of the points ti* in the subintervals. I don't like the way your text defined definite integral because we cannot compute the definite integral in the way that they indicate unless we know that the definite integral exists. That is, if we know the definite integral of f over [ a, b ] exists then we can compute it (or find it) using the limits in the book. However, just because one or the other of those limits exists this would not imply that the function was integrable. Your book skirts the issue by defining the definite integral for a continuous function-we can prove that if a function f is continuous function on an interval [ a, b ] ...

#### wk10calc14

Allan Hancock College, MAT 1102
Excerpt: ... Approximating Areas with Sums of Rectangles y y Definite Integrals If f is decreasing Lecture 14 Algebra & Calculus I t Left sum overestimates area. t Right sum underestimates area. Calculus: Definite Integrals Lect. 14 1 / 23 Calculus: Definite Integrals Lect. 14 4 / 23 Approximating Areas (Continued) y y Approximating Areas (Continued) y If f is constant If f is increasing t Left sum underestimates area. Right sum overestimates area. t Left sum = Right sum. t Calculus: Definite Integrals Lect. 14 5 / 23 Calculus: Definite Integrals Lect. 14 6 / 23 Approximating Areas (Continued) In every case, the exact area is between the left and right sums. Also, to get a better approximation, use narrower rectangles. y Matlab Notes Left, Right and Mid-point sums are all examples of Riemann sum s. Example Summing area under y = x2 from x = 0 to x = 1 using left, right, and mid-point sums. Which one is more accurate? (Exact area is 1/3.) % Approx area under y=x^2 from x=0 to x=1 h=0.1; % the ste ...

#### review1s

UC Davis, MATH 1
Excerpt: ... Review for Midterm 1 Information for Midterm 1: Where: Here in this classroom. When: This Friday (October 17,2008), on lecture time 12:101:00pm . Who: You, of course. What: Contents of Section 4.8 and Chapter 5. This is a closed book/notes/friends exam. Totally 4 pages, 6 problems. No calculator is allowed. A previous exam for the last year has been posted on the course webpage. You are encouraged to take it rst, and then compare your answers with the provided solutions. Helpful resources: This Review Previous Exam (posted on the course webpage) Homework; In particular, PAPER HOMEWORK! Lecture Notes Textbook Online learning materials. Some basic concepts/methods/theorems Antiderivatives; Indenite integral Initial value problem; Partition of an interval and the norm of a partition; First Prev Next Last Go Back Full Screen Close Quit Riemann Sum s. In particular, upper sum and lower sum Sigma notation Average of an ...

#### Lesson5.3RiemannSumsDefiniteIntegrals

LeTourneau, MATH 1903
Excerpt: ... Riemann Sum s and the Definite Integral Lesson 5.3 Why? Why is the area of the yellow rectangle at the end = x ( f (b) f (a) ) a x b Review f(x) a b We partition the interval into n sub-intervals ba x = n Evaluate f(x) at right endpoints a + k x of kth sub-interval for k = 1, 2, 3, n Review f(x) a n b Look at Java demo Sum S n = lim f (a + k x) x n k =1 We expect Sn to improve thus we define A, the area under the curve, to equal the above limit. Riemann Sum 1. Partition the interval [a,b] into n subintervals a = x0 < x1 < xn-1< xn = b Call this partition P The kth subinterval is xk = xk-1 xk Largest xk is called the norm, called |P| 2. Choose an arbitrary value from each subinterval, call it ci Riemann Sum 1. Form the sum Rn = f (c1 )x1 + f (c2 )x2 + . + f (cn )xn = f (ci )xi i =1 n This is the Riemann sum associated with the function f the given partitio ...