Numerical Integration Midpoint Integration. Numerical Integration The problem of finding a value for certain integrals is one of the first problems that.

Slides:

Advertisements

Similar presentations

Numerical Integration Romberg Extrapolation. Acceleration The term acceleration is a term sometimes used in numerical analysis that refers to how you.

Advertisements

MTH 252 Integral Calculus Chapter 8 – Principles of

Homework Homework Assignment #2 Read Section 5.3

8 TECHNIQUES OF INTEGRATION. There are two situations in which it is impossible to find the exact value of a definite integral. TECHNIQUES OF INTEGRATION.

1 Fundamental Theorem of Calculus Section The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F.

1 Example 2 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,  ] into 6 subintervals. Solution Observe that the function.

NUMERICAL DIFFERENTIATION The derivative of f (x) at x 0 is: An approximation to this is: for small values of h. Forward Difference Formula.

1 Example 1 (a) Estimate by the Midpoint, Trapezoid and Simpson's Rules using the regular partition P of the interval [0,2] into 6 subintervals. (b) Find.

5.2 Definite Integrals Quick Review Quick Review Solutions.

Numerical Integration Simpson’s Method. Trapezoid Method & Polygonal Lines One way to view the trapezoid method for integration is that the curves that.

Simulation of Random Walk How do we investigate this numerically? Choose the step length to be a=1 Use a computer to generate random numbers r i uniformly.

Numerical Computation

MAT 1235 Calculus II Section 7.7 Approximate (Numerical) Integration

4.6 Numerical Integration -Trapezoidal Rule -Simpson’s Rule

Lecture 19 – Numerical Integration 1 Area under curve led to Riemann sum which led to FTC. But not every function has a closed-form antiderivative.

Wicomico High School Mrs. J. A. Austin AP Calculus 1 AB Third Marking Term.

Trapezoidal Approximation Objective: To find area using trapezoids.

The Integral chapter 5 The Indefinite Integral Substitution The Definite Integral As a Sum The Definite Integral As Area The Definite Integral: The Fundamental.

THE DEFINITE INTEGRAL RECTANGULAR APPROXIMATION, RIEMANN SUM, AND INTEGRTION RULES.

Georg Friedrich Bernhard Riemann

Chapter 5 Integral. Estimating with Finite Sums Approach.

Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1]

A REA A PPROXIMATION 4-B. Exact Area Use geometric shapes such as rectangles, circles, trapezoids, triangles etc… rectangle triangle parallelogram.

1 5.e – The Definite Integral as a Limit of a Riemann Sum (Numerical Techniques for Evaluating Definite Integrals)

Section 5.6 Integration: “The Fundamental Theorem of Calculus”

7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.

1 §12.4 The Definite Integral The student will learn about the area under a curve defining the definite integral.

Learning Objectives for Section 13.4 The Definite Integral

Chapter 6 The Definite Integral. § 6.1 Antidifferentiation.

5.1.  When we use the midpoint rule or trapezoid rule we can actually calculate the maximum error in the calculation to get an idea how much we are off.

4.4 The Fundamental Theorem of Calculus If a function is continuous on the closed interval [a, b], then where F is any function that F’(x) = f(x) x in.

The Fundamental Theorems of Calculus Lesson 5.4. First Fundamental Theorem of Calculus Given f is  continuous on interval [a, b]  F is any function.

Chapter 5: The Definite Integral Section 5.2: Definite Integrals

The Fundamental Theorem of Calculus Objective: The use the Fundamental Theorem of Calculus to evaluate Definite Integrals.

Section 5.9 Approximate Integration Practice HW from Stewart Textbook (not to hand in) p. 421 # 3 – 15 odd.

Antidifferentiation: The Indefinite Intergral Chapter Five.

Math – Integration 1. We know how to calculate areas of some shapes. 2.

MAT 212 Brief Calculus Section 5.4 The Definite Integral.

The Fundamental Theorem of Calculus

Section 6.1 Antiderivatives Graphically and Numerically.

12 INFINITE SEQUENCES AND SERIES. In general, it is difficult to find the exact sum of a series.  We were able to accomplish this for geometric series.

Distance Traveled Area Under a curve Antiderivatives

Chapter 6 INTEGRATION An overview of the area problem The indefinite integral Integration by substitution The definition of area as a limit; sigma notation.

Antiderivatives and Indefinite Integration

In Chapters 6 and 8, we will see how to use the integral to solve problems concerning:  Volumes  Lengths of curves  Population predictions  Cardiac.

1 Example 1 Estimate by the six Rectangle Rules using the regular partition P of the interval [0,1] into 4 subintervals. Solution This definite integral.

6. Numerical Integration 6.1 Definition of numerical integration. 6.2 Reasons to use numerical integration. 6.3 Formulas of numerical Integration. 6.4.

Riemann Sums and The Definite Integral. time velocity After 4 seconds, the object has gone 12 feet. Consider an object moving at a constant rate of 3.

5.1 Approximating Area Thurs Feb 18 Do Now Evaluate the integral 1)

Chapter 6 Integration Section 4 The Definite Integral.

Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

Lecture 19 Approximating Integrals. Integral = “Area”

Clicker Question 1 What is ? (Hint: u-sub) – A. ln(x – 2) + C – B. x – x 2 + C – C. x + ln(x – 2) + C – D. x + 2 ln(x – 2) + C – E. 1 / (x – 2) 2 + C.

Definite Integrals & Riemann Sums

Definite Integrals, The Fundamental Theorem of Calculus Parts 1 and 2 And the Mean Value Theorem for Integrals.

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

INFINITE SEQUENCES AND SERIES In general, it is difficult to find the exact sum of a series.  We were able to accomplish this for geometric series and.

NUMERICAL DIFFERENTIATION Forward Difference Formula

4.4 The Fundamental Theorem of Calculus

Trapezoid Method and Iteration & Acceleration

Lecture 19 – Numerical Integration

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

5. 7a Numerical Integration. Trapezoidal sums

Composite Numerical Integration

Objective: Be able to approximate the area under a curve.

5. 7a Numerical Integration. Trapezoidal sums

Objective: Be able to approximate the area under a curve.

AP Calculus December 1, 2016 Mrs. Agnew

Section 4 The Definite Integral

Presentation transcript:

Numerical Integration Midpoint Integration

Numerical Integration The problem of finding a value for certain integrals is one of the first problems that was attempted by numerical methods. There are certain integrals whose values are necessary to be able to do basic calculations, but at the same time the integrals do not have a closed form antiderivative for the integrand. This does not allow the application of the Fundamental Theorem of Calculus (this is the way that you have gotten used to evaluating integrals). The classic example is the normal distribution curve in statistics. Terminology A partition (of size n ) of an interval [ a, b ] is a set of points: a = x 0 < x 1 < x 2 < x 3 < … < x n = b A regular partition has all the points equally spaced Each pair of consecutive points forms a subinterval [ x i, x i +1 ]. Midpoint Integration One of the easiest integration methods to implement (although not very efficient computationally) is the midpoint method of integration. This uses a series of rectangles to estimate the value of the integral with the height of each rectangle the function value evaluates at the midpoint of each subinterval.

a=x 0 x1x1 x2x2 x 3 =b f(x) The diagram to the right shows a regular partition of size three. The yellow, blue and green rectangles are the estimates for the midpoint method. Accuracy & Measuring Error As the number of points n in a regular partition increases the absolute error between the midpoint method estimate and the actual value of the integral decreases. We can form a sequence of estimates based on the number of points in a regular partition n, since the points in the partition are completely determined by the number n. Call the value of each midpoint estimate M n. Any of the measurements of errors in sequences can then be used, most commonly the standard Cauchy error.

Algorithm for Midpoint Method function intf ( a, b, n ) deltax = ( b - a )/ n mid = a + deltax /2 intsum = 0 for( i =1, i  n, i ++, intsum = intsum + f(mid) mid = mid + deltax ) intf = deltax * intsum prevint = intf (a, b, 2) nextint = intf (a, b, 3) for( i = 4, i  maximum iterations && |prevint – nextint|  error, i++, prevint = nextint nextint = intf ( a, b, i ) Formula for Midpoint Method with a partition of size n

Example Estimate the integral to the right with a partition of size 3 and a partition of size 4. Partition size 3 Partition size 4