Sec 5.1: Areas and Distances

Slides:

Advertisements

Similar presentations

Section 8.5 Riemann Sums and the Definite Integral.

Advertisements

Applying the well known formula:

INTEGRALS Areas and Distances INTEGRALS In this section, we will learn that: We get the same special type of limit in trying to find the area under.

Copyright © Cengage Learning. All rights reserved. 5 Integrals.

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.

5.2 Definite Integrals Quick Review Quick Review Solutions.

Riemann Sums and the Definite Integral Lesson 5.3.

Math 5A: Area Under a Curve. The Problem: Find the area of the region below the curve f(x) = x 2 +1 over the interval [0, 2].

Chapter 5 Integral. Estimating with Finite Sums Approach.

INTEGRALS Areas and Distances INTEGRALS In this section, we will learn that: We get the same special type of limit in trying to find the area under.

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

Introduction to integrals Integral, like limit and derivative, is another important concept in calculus Integral is the inverse of differentiation in some.

Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

Section 4.3 – Riemann Sums and Definite Integrals

INTEGRALS 5. INTEGRALS In Chapter 2, we used the tangent and velocity problems to introduce the derivative—the central idea in differential calculus.

Integrals  In Chapter 2, we used the tangent and velocity problems to introduce the derivative—the central idea in differential calculus.  In much the.

Copyright © Cengage Learning. All rights reserved. 4 Integrals.

Section 4.1 Areas and Distances Math 1231: Single-Variable Calculus.

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

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

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,

Sigma Notations Example This tells us to start with k=1 This tells us to end with k=100 This tells us to add. Formula.

Area of a Plane Region We know how to find the area inside many geometric shapes, like rectangles and triangles. We will now consider finding the area.

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

Discuss how you would find the area under this curve!

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

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

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

RIEMANN SUMS AP CALCULUS MS. BATTAGLIA. Find the area under the curve from x = 0 to x = 35. The graph of g consists of two straight lines and a semicircle.

5.1 Approximating Area Thurs Feb 18 Do Now Evaluate the integral 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,

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

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

Definite Integrals & Riemann Sums

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.

DO NOW: v(t) = e sint cost, 0 ≤t≤2∏ (a) Determine when the particle is moving to the right, to the left, and stopped. (b) Find the particles displacement.

Copyright © Cengage Learning. All rights reserved. 4 Integrals.

INTEGRALS 5. INTEGRALS In Chapter 3, we used the tangent and velocity problems to introduce the derivative—the central idea in differential calculus.

[5-4] Riemann Sums and the Definition of Definite Integral Yiwei Gong Cathy Shin.

Copyright © Cengage Learning. All rights reserved.

Chapter 5 Integrals 5.1 Areas and Distances

Copyright © Cengage Learning. All rights reserved.

Riemann Sums and the Definite Integral

Area and the Definite Integral

Riemann Sums as Estimates for Definite Integrals

Riemann Sums and the Definite Integral

5.1 – Estimating with Finite Sums

5. 7a Numerical Integration. Trapezoidal sums

Section 5.1: Estimating with Finite Sums

Area and the Definite Integral

Area & Riemann Sums Chapter 5.1

Applying the well known formula:

5. 7a Numerical Integration. Trapezoidal sums

Sec 5.1: Areas and Distances

Copyright © Cengage Learning. All rights reserved.

Find the general indefinite integral. {image}

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

§ 6.2 Areas and Riemann Sums.

Riemann Sums as Estimates for Definite Integrals

Area Under a Curve Riemann Sums.

5.1 Areas and Distances Approximating the area under a curve with rectangles or trapezoids, and then trying to improve our approximation by taking.

4.2 – Areas 4.3 – Riemann Sums Roshan Roshan.

6.1 Estimating with Finite Sums

Areas and Distances In this handout: The Area problem

Sec 5.1: Areas and Distances

Presentation transcript:

Sec 5.1: Areas and Distances

Sec 5.1: Areas and Distances

Sec 5.1: Areas and Distances The Area Problem Divide it into triangles it isn’t so easy to find the area of a region with curved sides

Sec 5.1: Areas and Distances Find the area of the region that lies under the curve from to Use two rectangles and left endpoints

Sec 5.1: Areas and Distances

Sec 5.1: Areas and Distances Two rectangles with left endpoint 4 rectangles with left endpoint Example: Use rectangles to estimate the area under the parabola from 0 to 1 4 rectangles with right endpoint 4 rectangles with midpoint

Sec 5.1: Areas and Distances 4 rectangles with right endpoint 4 rectangles with midpoint 4 rectangles with left endpoint Estimating area with 4 rectangles using left end points Estimating area with 4 rectangles using right end points Estimating area with 4 rectangles using midpoints Notations:

Sec 5.1: Areas and Distances

Sec 5.1: Areas and Distances Math-101 Final Exam Term-131 apply the idea to problem with large number of rectangles or more general problem (n rectangles)

Example: Sec 5.1: Areas and Distances Use rectangles to estimate the area under the parabola from 0 to 1 We could obtain better estimates by increasing the number of rectangles

Sec 5.1: Areas and Distances Let’s apply the idea to the more general problem We start by subdividing the interval [a,b] into n subintervals The width of the interval [a,b] is b-a the width of each subinterval is The subintervals are

1 2 3 Sec 5.1: Areas and Distances Step Step Step partition Riemann sum for ƒ on the interval [a, b].

Sec 5.1: Areas and Distances Math-101 Final Exam Term-131

Sec 5.1: Areas and Distances Term-091

Sec 5.1: Areas and Distances Term-103

Remark: Sec 5.1: Areas and Distances Actual area How to get actual area ?

Remark: Sec 5.1: Areas and Distances Actual area Actual area Big triangle _ Small triangel

Note: Area under a curve = limit of summation Sec 5.1: Areas and Distances Note: Area under a curve = limit of summation Note: Area under the curve equals 4

Note: Area under a curve = limit of summation Sec 5.1: Areas and Distances Note: Area under a curve = limit of summation Note:

Sec 5.1: Areas and Distances Example: Use rectangles to estimate the area under the parabola from 0 to 1

Sec 5.1: Areas and Distances EXAM-1 TERM-102

Sec 5.1: Areas and Distances Example: Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval. We take speedometer readings every five seconds and record them in the following table: Distance = velocity X time assume that the velocity is constant in every 5-second interval estimate for the total distance traveled: 1135 ft