5.5 The Trapezoidal Rule. I. Trapezoidal Rule A.) Area of a Trapezoid -

Slides:

Advertisements

Similar presentations

Section 7.6 – Numerical Integration

Advertisements

Adguary Calwile Laura Rogers Autrey~ 2nd Per. 3/14/11

Section 8.5 Riemann Sums and the Definite Integral.

Section 5.5. All of our RAM techniques utilized rectangles to approximate areas under curves. Another geometric shape may do this job more efficiently.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Trapezoidal Rule Section 5.5.

Applying the well known formula:

APPLICATIONS OF INTEGRATION

Ch 5.5 Trapezoidal Rule Graphical, Numerical, Algebraic by Finney Demana, Waits, Kennedy.

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 Greg Kelly, Hanford High School, Richland, Washington.

Riemann Sums and the Definite Integral Lesson 5.3.

Integration. Problem: An experiment has measured the number of particles entering a counter per unit time dN(t)/dt, as a function of time. The problem.

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.

Chapter 5 Integral. Estimating with Finite Sums Approach.

Distance, Displacement, and Acceleration for Linear Motion Lesson 10-2.

Chapter 6 The Definite Integral. § 6.1 Antidifferentiation.

3. Fundamental Theorem of Calculus. Fundamental Theorem of Calculus We’ve learned two different branches of calculus so far: differentiation and integration.

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

SECTION 5.1: ESTIMATING WITH FINITE SUMS Objectives: Students will be able to… Find distance traveled Estimate using Rectangular Approximation Method Estimate.

Chapter 5 – The Definite Integral. 5.1 Estimating with Finite Sums Example Finding Distance Traveled when Velocity Varies.

1/12/2016HL Math - Santowski1 Lesson 50 – Area Between Curves HL Math- Santowski.

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

Section 3.2 – Calculating Areas; Riemann Sums

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

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.

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

AP CALC: CHAPTER 5 THE BEGINNING OF INTEGRAL FUN….

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

Chapter Definite Integrals Obj: find area using definite integrals.

Warm up Problems 1) Approximate using left- and right- hand Riemann sums, n = 6. 2) 3) Find without a calculator.

Lecture 19 Approximating Integrals. Integral = “Area”

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

Area under a curve To the x axis.

9/26/2016HL Math - Santowski1 Lesson 49 – Working with Definite Integrals HL Math- Santowski.

Chapter 5 AP Calculus BC.

Applications of RAM Section 5.1b.

Approximate Integration

Riemann Sums and the Definite Integral

Riemann Sums as Estimates for Definite Integrals

Copyright © Cengage Learning. All rights reserved.

Review Practice problems

Riemann Sums and the Definite Integral

5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.

Integration Review Problems

Section 5.1: Estimating with Finite Sums

Lesson 48 – Area Between Curves

Section 3.2 – Calculating Areas; Riemann Sums

Unit 6 – Fundamentals of Calculus Section 6

Lesson 46 – Working with Definite Integrals

Applying the well known formula:

Lesson 5-R Review of Chapter 5.

Fundamental Theorem of Calculus

Ch. 6 – The Definite Integral

Copyright © Cengage Learning. All rights reserved.

A car is decelerated to 20 m/s in 6 seconds

Section 3.2 – Calculating Areas; Riemann Sums

4.2/4.6 Approximating Area Mt. Shasta, California.

5.1 Calculus and Area.

AREA Section 4.2.

True or False: The exact length of the parametric curve {image} is {image}

Numerical Integration

6.2 Definite Integrals.

Drill 80(5) = 400 miles 48 (3) = 144 miles a(t) = 10

Objectives Approximate a definite integral using the Trapezoidal Rule.

Riemann Sums as Estimates for Definite Integrals

Copyright © Cengage Learning. All rights reserved.

True or False: The exact length of the parametric curve {image} is {image}

Sec 5.1: Areas and Distances

Presentation transcript:

5.5 The Trapezoidal Rule

I. Trapezoidal Rule A.) Area of a Trapezoid -

B.) Approximate the area under the following curve from x = 1 to x = 4 using 6 subintervals of trapezoids.

VISUALLY:

TRAM:

TRAM:

C. Theorem: The Trapezoidal Rule To approximateusing trapezoids Where [a, b] is partitioned into n subintervals equal length

D. Using the Calculator:

II. Examples 1. Use 5 trapezoids to approximate the area bounded by the curve of, the x- axis, and the vertical lines x = 0 and x = 5. Write out each term of the summation and confirm on your calculator.

Visually:

2. The table shows the velocity of a car traveling on a highway at different times. Use TRAM to estimate the total distance traveled over the time interval. t(sec) v(t) (ft/sec)

3. The table shows the reading of outdoor temperatures from noon to midnight for a certain day. Estimate the average temperature for the 12 hour interval using the trapezoidal rule. Time12 PM AM Temp

II. Simpson’s Rule Given a nonnegative function f (x), i.e., f (x) > 0, on [a, b]. Find the area bounded by the curve, the x- axis, and the vertical lines x = a and x = b.

II. Simpson’s Rule Given a nonnegative function f (x), i.e., f (x) > 0, on [a, b]. Find the area bounded by the curve, the x- axis, and the vertical lines x = a and x = b.