LarPCalcLim2_12_05

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Slide 1:

Copyright © Cengage Learning. All rights reserved. 12 Limits and an Introduction to Calculus

Slide 2:

12.5 Copyright © Cengage Learning. All rights reserved. THE AREA PROBLEM

What You Should Learn:

Find limits of summations. Use rectangles to approximate areas of plane regions. Use limits of summations to find areas of plane regions. What You Should Learn

Slide 4:

Limits of Summations

Limits of Summations:

Limits of Summations We have used the concept of a limit to obtain a formula for the sum S of an infinite geometric series Using limit notation, this sum can be written as

Limits of Summations:

Limits of Summations The following summation formulas and properties are used to evaluate finite and infinite summations.

Example 1 – Evaluating a Summation:

Example 1 – Evaluating a Summation Evaluate the summation. Solution: Using the second summation formula with n = 200, you can write

Example 1 – Solution:

Example 1 – Solution . cont’d

Slide 9:

The Area Problem

The Area Problem:

The Area Problem You now have the tools needed to solve the second basic problem of calculus: the area problem. The problem is to find the area of the region R bounded by the graph of a nonnegative, continuous function f , the x -axis, and the vertical lines x = a and x = b , as shown in Figure 12.33. Figure 12.33

The Area Problem:

The Area Problem If the region R is a square, a triangle, a trapezoid, or a semicircle, you can find its area by using a geometric formula. For more general regions, however, you must use a different approach—one that involves the limit of a summation. The basic strategy is to use a collection of rectangles of equal width that approximates the region R , as illustrated in Example 4.

Example 4 – Approximating the Area of a Region:

Example 4 – Approximating the Area of a Region Use the five rectangles in Figure 12.34 to approximate the area of the region bounded by the graph of f ( x ) = 6 – x 2 , the x -axis, and the lines x = 0 and x = 2. Figure 12.34

Example 4 – Solution:

Example 4 – Solution Because the length of the interval along the x -axis is 2 and there are five rectangles, the width of each rectangle is . The height of each rectangle can be obtained by evaluating f at the right endpoint of each interval. The five intervals are as follows. Notice that the right endpoint of each interval is for i = 1, 2, 3, 4, 5.

Example 4 – Solution:

Example 4 – Solution The sum of the areas of the five rectangles is cont’d Height Width

Example 4 – Solution:

Example 4 – Solution So, you can approximate the area of R as 8.48 square units. cont’d

The Area Problem:

The Area Problem By increasing the number of rectangles used in Example 4, you can obtain closer and closer approximations of the area of the region. For instance, using 25 rectangles of width each, you can approximate the area to be A  9.17 square units. The following table shows even better approximations.

The Area Problem:

The Area Problem Based on the procedure illustrated in Example 4, the exact area of a plane region R is given by the limit of the sum of n rectangles as n approaches .

Example 5 – Finding the Area of a Region:

Example 5 – Finding the Area of a Region Find the area of the region bounded by the graph of f ( x ) = x 2 and the x -axis between x = 0 and x = 1, as shown in Figure 12.35. Figure 12.35

Example 5 – Solution:

Example 5 – Solution Begin by finding the dimensions of the rectangles. Width: Height:

Example 5 – Solution:

Example 5 – Solution Next, approximate the area as the sum of the areas of n rectangles. cont’d Summation form

Example 5 – Solution:

Example 5 – Solution Finally, find the exact area by taking the limit as n approaches . cont’d Rational form

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