2/28/2016 Perkins AP Calculus AB Day 15 Section 4.6.

Slides:

Advertisements

Similar presentations

Numerical Integration

Advertisements

6. 4 Integration with tables and computer algebra systems 6

We sometimes need an efficient method to estimate area when we can not find the antiderivative.

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.

5/16/2015 Perkins AP Calculus AB Day 5 Section 4.2.

Riemann Sums Jim Wang Mr. Brose Period 6. Approximate the Area under y = x² on [ 0,4 ] a)4 rectangles whose height is given using the left endpoint b)4.

Applying the well known formula:

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.

5.5 Numerical Integration Mt. Shasta, California Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1998.

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.

Trapezoidal Approximation Objective: To find area using trapezoids.

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

Chapter 3: Derivatives Section 3.3: Rules for Differentiation

1 Copyright © 2015, 2011 Pearson Education, Inc. Chapter 5 Integration.

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.

In this section, we will investigate how to estimate the value of a definite integral when geometry fails us. We will also construct the formal definition.

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.

4-6Trapezoidal Rule: A Mathematics Academy Production.

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.

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.

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

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,

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

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.

AP CALCULUS AB CHAPTER 4, SECTION 2(ISH) Area 2013 – 2014 Revised

Section 4.3 Day 2 Riemann Sums & Definite Integrals AP Calculus BC.

SECTION 4.2: AREA AP Calculus BC. LEARNING TARGETS: Use Sigma Notation to evaluate a sum Apply area formulas from geometry to determine the area under.

Definite Integrals & Riemann Sums

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

7.2: Riemann Sums: Left & Right-hand Sums

Application of the Integral

5.5 The Trapezoid Rule.

Lecture 19 – Numerical Integration

Riemann Sums and the Definite Integral

Section 4-2-B More approximations of area using

Clicker Question 1 What is ? A. x tan(x2) + C

Trapezoidal Approximation

Midpoint and Trapezoidal Rules

More Approximations Left-hand sum: use rectangles, left endpoint used for height Right-hand sum: use rectangles, right endpoint used for height Midpoint.

NUMERICAL INTEGRATION

Riemann Sums Approximate area using rectangles

5. 7a Numerical Integration. Trapezoidal sums

Approximating Definite Integrals. Left Hand Riemann Sums.

Approximating Definite Integrals. Left Hand Riemann Sums.

Integration & Area Under a Curve

Limits of Riemann’s Sum

Applications of Integration

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

5.5 Numerical Integration

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

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

Warm-up 2/4 Find the AVERAGE VALUE of

4.3 Day 1 Exit Ticket for Feedback

Estimating with Finite Sums

5.5 Numerical Integration

4.2/4.6 Approximating Area Mt. Shasta, California.

Section 4.2A Calculus AP/Dual, Revised ©2018

5.5 Numerical Integration

AP Calculus December 1, 2016 Mrs. Agnew

Area Under a Curve Riemann Sums.

Chapter 5 Integration.

Jim Wang Mr. Brose Period 6

Presentation transcript:

2/28/2016 Perkins AP Calculus AB Day 15 Section 4.6

Trapezoidal Rule Approximates the area under a curve using trapezoids instead of rectangles. where and both are y-values. When using trapezoidal rule to find the area, you’ll use each endpoint once and each multiple of twice.

Approximate using the trapezoidal rule and n = 4. Actual area:

Estimate if n = 3 using the left, right, midpoint and trapezoidal sums. Left Sum: Right Sum: Midpoint Sum: Trapezoidal Sum:

Perkins AP Calculus AB Day 15 Section 4.6

Trapezoidal Rule Approximates the area under a curve using trapezoids instead of rectangles.

Approximate using the trapezoidal rule and n = 4.

Estimate if n = 3 using the left, right, midpoint and trapezoidal sums. Left Sum: Right Sum: Midpoint Sum: Trapezoidal Sum: