Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
See Example 1 in Section 5.1. Now suppose that the region S is divided into 5 strips of equal width. Let each strip be approximated with a rectangle whose height is determined using a right endpoint. a) Find the sum of the areas of these 5 rectangles. R b) Which of the following is true? O R, underestimates the area of S O R, overestimates the area of S OR, equals the area of S

See Example 1 in Section 5.1. Now suppose that the region S is divided into 5 strips of equal width. Let each strip be approximated with a rectangle whose height is determined using a right endpoint. a) Find the sum of the areas of these 5 rectangles. R b) Which of the following is true? O R, underestimates the area of S O R, overestimates the area of S OR, equals the area of S

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.3: Quadratic Equations
Problem 78E
See similar textbooks
icon
Related questions
Topic Video
Question

I need help with this question.

 

See Example 1 in Section 5.1. Now suppose that the region S is divided into 5 strips of equal width. Let each strip be approximated with a rectangle whose height is determined using a right endpoint. a) Find the sum of the areas of these 5 rectangles. R3 = b) Which of the following is true? O R, underestimates the area of S O R, overestimates the area of S O R, equals the area of S Question Help: O Message instructor D Post to forum Submit Question
Transcribed Image Text:See Example 1 in Section 5.1. Now suppose that the region S is divided into 5 strips of equal width. Let each strip be approximated with a rectangle whose height is determined using a right endpoint. a) Find the sum of the areas of these 5 rectangles. R3 = b) Which of the following is true? O R, underestimates the area of S O R, overestimates the area of S O R, equals the area of S Question Help: O Message instructor D Post to forum Submit Question
V EXAMPLE 1 Estimating an area Use rectangles to estimate the area under the parabola y = x' from 0 to 1 (the parabolic region S illustrated in Figure 3). SOLUTION We first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips Sj, S2, S, and S, by drawing the vertical lines x = ;. x = , and x = as in Figure 4(a). FIGURE 3 у f1,1) (1,1) y = S, S, S, S, FIGURE 4 (a) (b) 333 SECTION 5.1 AREAS AND DISTANCE! we can approximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip [see Figure 4(h)]. In other words, the heights of these rectangles are the values of the function (() = x' at the right end- points of the subintervals (0. J. [. }|. (:. :), and [;. 1]. Each rectangle has width and the heights are (}, (. (GY, and 1'. If we let R. be the sum of the areas of these approximating rectangles, we get R, = · (:)} + } · (C) + ! · G}' + ! • 1² = ; = 0,46875 From Figure 4(b) we see that the area A of S is less than Ra, so A< 0.46875 Instead of using the rectangles in Figure 4(b) we could use the smaller rectangles in Figure 5 whose heights are the values of f at the left endpoints of the subintervals. (The lertmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is y =r L, = : 0° +: (' + : G} +! · () = = 0.21875 We see that the area of S is larger than L.. so we have lower and upper estimates for A: 0.21875 <A < 0.46875 We can repeat this procedure with a larger number of strips. Figure 6 shows what happens when we divide the region S into eight strips of equal width. FIGURE 5
Transcribed Image Text:V EXAMPLE 1 Estimating an area Use rectangles to estimate the area under the parabola y = x' from 0 to 1 (the parabolic region S illustrated in Figure 3). SOLUTION We first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips Sj, S2, S, and S, by drawing the vertical lines x = ;. x = , and x = as in Figure 4(a). FIGURE 3 у f1,1) (1,1) y = S, S, S, S, FIGURE 4 (a) (b) 333 SECTION 5.1 AREAS AND DISTANCE! we can approximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip [see Figure 4(h)]. In other words, the heights of these rectangles are the values of the function (() = x' at the right end- points of the subintervals (0. J. [. }|. (:. :), and [;. 1]. Each rectangle has width and the heights are (}, (. (GY, and 1'. If we let R. be the sum of the areas of these approximating rectangles, we get R, = · (:)} + } · (C) + ! · G}' + ! • 1² = ; = 0,46875 From Figure 4(b) we see that the area A of S is less than Ra, so A< 0.46875 Instead of using the rectangles in Figure 4(b) we could use the smaller rectangles in Figure 5 whose heights are the values of f at the left endpoints of the subintervals. (The lertmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is y =r L, = : 0° +: (' + : G} +! · () = = 0.21875 We see that the area of S is larger than L.. so we have lower and upper estimates for A: 0.21875 <A < 0.46875 We can repeat this procedure with a larger number of strips. Figure 6 shows what happens when we divide the region S into eight strips of equal width. FIGURE 5
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 7 steps with 5 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Research Design Formulation
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
Geometry
ISBN:
9781337614085
Author:
Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:
Cengage,
College Algebra
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS