Numerical Approximations of Definite Integrals Mika Seppälä.

Slides:

Advertisements

Similar presentations

Solved Problems on Numerical Integration. Integration/Integration Techniques/Solved Problems on Numerical Integration by M. Seppälä Review of the Subject.

Advertisements

Numerical Integration

Volumes by Cylindrical Shells r. Integration/First Applications of Integration/Volumes by Cylindrical Shells by M. Seppälä Cylindrical Shells The method.

6. 4 Integration with tables and computer algebra systems 6

Section 8.5 Riemann Sums and the Definite Integral.

Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ.

INTEGRALS 5. INTEGRALS We saw in Section 5.1 that a limit of the form arises when we compute an area.  We also saw that it arises when we try to find.

16 MULTIPLE INTEGRALS.

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.

Area Between Two Curves

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.

Area Between Two Curves

Index FAQ Numerical Approximations of Definite Integrals Riemann Sums Numerical Approximation of Definite Integrals Formulae for Approximations Properties.

Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:

MAT 1235 Calculus II Section 7.7 Approximate (Numerical) Integration

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.

4.6 Numerical Integration Trapezoid and Simpson’s Rules.

4.6 Numerical Integration. The Trapezoidal Rule One method to approximate a definite integral is to use n trapezoids.

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)

Learning Objectives for Section 13.4 The Definite Integral

Chapter 5-The Integral Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

1 5.2 – The Definite Integral. 2 Review Evaluate.

CHAPTER Continuity The Definite Integral animation  i=1 n f (x i * )  x f (x) xx Riemann Sum xi*xi* xixi x i+1.

Definite Integrals Riemann Sums and Trapezoidal Rule.

Section 5.1/5.2: Areas and Distances – the Definite Integral Practice HW from Stewart Textbook (not to hand in) p. 352 # 3, 5, 9 p. 364 # 1, 3, 9-15 odd,

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

Chapter 15 – Multiple Integrals 15.1 Double Integrals over Rectangles 1 Objectives:  Use double integrals to find volumes  Use double integrals to find.

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.

Estimating area under a curve

INTEGRALS We saw in Section 5.1 that a limit of the form arises when we compute an area. We also saw that it arises when we try to find the distance traveled.

1. Does: ? 2. What is: ? Think about:. Finding Area between a Function & the x-axis Chapters 5.1 & 5.2 January 25, 2007.

SECTION 4-2-B More area approximations. Approximating Area using the midpoints of rectangles.

Observations about Error in Integral Approximations The Simplest Geometry.

Chapter 6 Integration Section 4 The Definite Integral.

Lesson 7-7 Numerical Approximations of Integrals -- What we do when we can’t integrate a function Riemann Sums Trapezoidal Rule.

Integration Review Part I When you see the words… This is what you think of doing…  A Riemann Sum equivalent to the definite integral is… -- 1.

Riemann Sums and Definite Integration y = 6 y = x ex: Estimate the area under the curve y = x from x = 0 to 3 using 3 subintervals and right endpoints,

Section 4.3 Day 1 Riemann Sums and Definite Integrals AP Calculus BC.

Chapter Definite Integrals Obj: find area using definite integrals.

Section 5.6: Approximating Sums Suppose we want to compute the signed area of a region bounded by the graph of a function from x = a to x = b.

Definite Integral df. f continuous function on [a,b]. Divide [a,b] into n equal subintervals of width Let be a sample point. Then the definite integral.

Lecture 19 Approximating Integrals. Integral = “Area”

Properties of Integrals. Mika Seppälä: Properties of Integrals Basic Properties of Integrals Through this section we assume that all functions are continuous.

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.

Area/Sigma Notation Objective: To define area for plane regions with curvilinear boundaries. To use Sigma Notation to find areas.

Copyright © Cengage Learning. All rights reserved. 7 Techniques of Integration.

5.2 – The Definite Integral. Introduction Recall from the last section: Compute an area Try to find the distance traveled by an object.

The Definite Integral. Area below function in the interval. Divide [0,2] into 4 equal subintervals Left Rectangles.

Approximating Antiderivatives. Can we integrate all continuous functions? Most of the functions that we have been dealing with are what are called elementary.

Riemann Sums A Method For Approximating the Areas of Irregular Regions.

NUMERICAL DIFFERENTIATION Forward Difference Formula

Lecture 19 – Numerical Integration

Riemann Sums and the Definite Integral

Section 4-2-B More approximations of area using

Midpoint and Trapezoidal Rules

NUMERICAL INTEGRATION

5. 7a Numerical Integration. Trapezoidal sums

Approximating Definite Integrals. Left Hand Riemann Sums.

Approximating Definite Integrals. Left Hand Riemann Sums.

Find an approximation to {image} Use a double Riemann sum with m = n = 2 and the sample point in the lower left corner to approximate the double integral,

TECHNIQUES OF INTEGRATION

A Method For Approximating the Areas of Irregular Regions

Applications of Integration

5. 7a Numerical Integration. Trapezoidal sums

Copyright © Cengage Learning. All rights reserved.

Objectives Approximate a definite integral using the Trapezoidal Rule.

§ 6.2 Areas and Riemann Sums.

Presentation transcript:

Numerical Approximations of Definite Integrals Mika Seppälä

Mika Seppälä: Numerical Integration Riemann Sums The definite integral of a positive function f over an interval [a,b] has been defined by Riemann sums which approximate the area under the graph of f. Taking more division points in the Riemann sums, the approximation of the area of the domain under the graph of f becomes better.

Mika Seppälä: Numerical Integration Definite Integrals This definition assumes that the limit does not depend on the various choices in the definition of the Riemann sums.

Mika Seppälä: Numerical Integration Numerical Approximations of Definite Integrals In view of the definition of the definite integral we may approximate its value by choosing the decomposition D to be a decomposition of the interval [a,b] into subintervals of length (b-a)/n for some positive integer n. The points  j can be freely chosen according to any rule from the intervals [x j-1,x j ]. In left rule approximations,  j =x j-1. In mid rule approximations,  j =(x j-1 + x j )/2. In right rule approximations,  j =x j.

Mika Seppälä: Numerical Integration Formulae for Approximations

Mika Seppälä: Numerical Integration Trapezoidal approximations and Simpson’s Formula Depending on the shape of the function in question, the following approximations are usually better: Trapezoidal Approximation: TRAP(n) = (LEFT(n)+RIGHT(n))/2 Simpson’s Approximation: SIMPSON(n)=(2MID(n)+TRAP(n))/3.

Mika Seppälä: Numerical Integration Properties of Approximations If f is strictly increasing – like in the above picture – then the above inequalities are also strict. If f is decreasing, then the direction of the above inequalities must be changed.

Mika Seppälä: Numerical Integration Properties of Approximations These estimates show that the approximations can be made as precise as needed simply by increasing the number n of subintervals.

Mika Seppälä: Numerical Integration Properties of Midpoint Approximations A function which is concave up has the property that its graph lies above any tangent line. This observation leads to the following estimate valid for functions that are concave up. The blue triangle on the right has been obtained by letting the top side of the rectangle on the left turn around the point where it intersects the graph of the function f. Since this is also the midpoint of the top side, the areas of the two blue domains are the same.