Arc Length … x y a b xi ... Pi P0 P1 Pn

Slides:

Advertisements

Similar presentations

Numerical Integration

Advertisements

6. 4 Integration with tables and computer algebra systems 6

Section 8.5 Riemann Sums and the Definite Integral.

A Riemann sum is a method for approximating the total area underneath a curve on a graph. This method is also known as taking an integral. There are 3.

Section 5.7 Numerical Integration. Approximations for integrals: Riemann Sums, Trapezoidal Rule, Simpson's Rule Riemann Sum: Trapezoidal Rule: Simpson’s.

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.

Approximate Integration: The Trapezoidal Rule Claus Schubert May 25, 2000.

1 Simpson’s 1/3 rd Rule of Integration. 2 What is Integration? Integration The process of measuring the area under a curve. Where: f(x) is the integrand.

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.

Chapter 7 – Techniques of Integration

4.6 Numerical Integration Trapezoid and Simpson’s Rules.

 Finding area of polygonal regions can be accomplished using area formulas for rectangles and triangles.  Finding area bounded by a curve is more challenging.

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

Homework questions thus far??? Section 4.10? 5.1? 5.2?

Section 4.3 – Riemann Sums and Definite Integrals

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.

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.

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

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.

Section 7.6 – Numerical Integration. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.

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

Section 7.6 – Numerical Integration. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.

5.5 Numerical Integration. concave down concave up concave down concave up concave down.

To find the area under the curve Warm-Up: Graph. Area under a curve for [0, 3]  The area between the x-axis and the function Warm-up What is the area.

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 the Definite Integral. represents the area between the curve 3/x and the x-axis from x = 4 to x = 8.

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,

Chapter Definite Integrals Obj: find area using definite integrals.

WS: Riemann Sums. TEST TOPICS: Area and Definite Integration Find area under a curve by the limit definition. Given a picture, set up an integral to calculate.

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.

Riemann Sum. Formula Step 1 Step 2 Step 3 Riemann Sum.

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.

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.

Numerical Integration using Trapezoidal Rule

NUMERICAL DIFFERENTIATION Forward Difference Formula

5.5 The Trapezoid Rule.

Lecture 19 – Numerical Integration

Approximate Integration

Riemann Sums and the Definite Integral

Romberg Rule Midpoint.

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.

The Area Question and the Integral

Integration & Area Under a Curve

TECHNIQUES OF INTEGRATION

Introduction to Integration

Applications of Integration

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

Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated.

5. 7a Numerical Integration. Trapezoidal sums

Ch. 6 – The Definite Integral

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

Warm-up 2/4 Find the AVERAGE VALUE of

Objectives Approximate a definite integral using the Trapezoidal Rule.

Area Under a Curve Riemann Sums.

Chapter 5 Integration.

Sec 5.1: Areas and Distances

Presentation transcript:

Arc Length … x y a b xi ... Pi P0 P1 Pn a b xi ... Pi P0 P1 Pn Curve C defined by y=f(x) – continuous on [a,b]

Length L of C: If f(x) – continuous, then by Mean Value Th,

Approximate integration Divide an interval [a,b] into n subintervals of width x, then the integral can be approximated by Riemann sum: Left end point approximation Right end point approximation Midpoint Rule Average of Left and Right end point approximation  Trapezoidal Rule

Error = amount that has to be added in order to make an approximation exact. Ln Rn Tn Mn 5 0.745635 0.645635 0.695635 0.691908 10 0.718771 0.668771 0.693771 0.692835 20 0.705803 0.680803 0.693303 0.693069 n EL ER ET EM 5 -0.052488 0.047512 -0.002488 0.001239 10 -0.025624 0.024376 -0.000624 0.000312 20 -0.012656 0.012344 -0.000156 0.000078 Error Bounds:

Approximation by parabola on each segment [xi,xi+2] (n - even)  a b xi P2 P0 P1 Pn xi+2 xi+1 Simpson’s Rule Error Bound: