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 _ ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...