Single Variable Calculus: Concepts and Contexts, Enhanced Edition

4th Edition

ISBN: 9781337687805

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter 5.2, Problem 6E

To determine

To estimate the right endpoints.

Expert Solution

Answer to Problem 6E

The value of right endpoints is −0.5 .

Explanation of Solution

Given information:

The equation is

∫−33g(x)dx with n=6

Calculation:

The right endpoints can be estimate as

∫−33g(x)dx

The endpoints of the 6 subintervals are [−3,−2],[−2,−1],[−1,0],[0,1],[1,2],[2,3]

So, the right endpoints are −2,−1,0,1,2,3

The width of the subintervals is

Δx=b−anΔx=3−(−3)6 [since, b=3,a=−3,n=6]Δx=3+36Δx=66Δx=1 [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(−2)=1g(−1)=−0.5g(0)=−1.5g(1)=−1.5g(2)=−0.5g(3)=2.5

Therefore,

The right endpoint rule gives

∫−33g(x)dx=Δx[g(−2)+g(−1)+g(0)+g(1)+g(2)+g(3)]

Put the values of Δx=1,g(−2)=1,g(−1)=−0.5,g(0)=−1.5,g(1)=−1.5,g(2)=−0.5 and g(3)=2.5 on the above equation

∫−33g(x)dx=Δx[g(−2)+g(−1)+g(0)+g(1)+g(2)+g(3)] =1(1+(−0.5)+(−1.5)+(−1.5)+(−0.5)+2.5) =1(1−0.5−1.5−1.5−0.5+2.5) =1(3.5−4) =1(−0.5) =−0.5

Hence,

The value of right endpoints is −0.5 .

To determine

To estimate the left endpoints.

Expert Solution

Answer to Problem 6E

The value of left endpoints is −1 .

Explanation of Solution

Given information:

The equation is

∫−33g(x)dx with n=6

Calculation:

The left endpoints can be estimate as

∫−33g(x)dx

The endpoints of the 6 subintervals are [−3,−2],[−2,−1],[−1,0],[0,1],[1,2],[2,3]

So, the left endpoints are −3,−2,−1,0,1,2

The width of the subintervals is

Δx=b−anΔx=3−(−3)6 [since, b=3,a=−3,n=6]Δx=3+36Δx=66Δx=1 [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(−3)=2g(−2)=1g(−1)=−0.5g(0)=−1.5g(1)=−1.5g(2)=−0.5

Therefore,

The right endpoint rule gives

∫−33g(x)dx=Δx[g(−3)+g(−2)+g(−1)+g(0)+g(1)+g(2)]

Put the values of Δx=1,g(−3)=2,g(−2)=1,g(−1)=−0.5,g(0)=−1.5,g(1)=−1.5 and g(2)=−0.5 on the above equation

∫−33g(x)dx=Δx[g(−3)+g(−2)+g(−1)+g(0)+g(1)+g(2)] =1(2+1+(−0.5)+(−1.5)+(−1.5)+(−0.5)) =1(2+1−0.5−1.5−1.5−0.5) =1(3−4) =1(−1) =−1

Hence,

The value of left endpoints is −1 .

To determine

To estimate the midpoints.

Expert Solution

Answer to Problem 6E

The value of midpoints is −1.75 .

Explanation of Solution

Given information:

The equation is

∫−33g(x)dx with n=6

Calculation:

The midpoints can be estimate as

∫−33g(x)dx

The endpoints of the 6 subintervals are [−3,−2],[−2,−1],[−1,0],[0,1],[1,2],[2,3]

So, the midpoints are −2.5,−1.5,−0.5,0.5,1.5,2.5

The width of the subintervals is

Δx=b−anΔx=3−(−3)6 [since, b=3,a=−3,n=6]Δx=3+36Δx=66Δx=1 [divede numerator and denominator by 6]

From the given graph in the question we can observed that

g(−2.5)=1.5g(−1.5)=0g(−0.5)=−1g(0.5)=−1.75g(1.5)=−1g(2.5)=0.5

Therefore,

The midpoint rule gives

∫−33g(x)dx=Δx[g(−2.5)+g(−1.5)+g(−0.5)+g(0.5)+g(1.5)+g(2.5)]

Put the values of Δx=1,g(−2.5)=1.5,g(−1.5)=0.5,g(−0.5)=−1,g(0.5)=−1.75,g(1.5)=−1 and g(2.5)=0.5 on the above equation

∫−33g(x)dx=Δx[g(−2.5)+g(−1.5)+g(−0.5)+g(0.5)+g(1.5)+g(2.5)] =1(1.5+0+(−1)+(−1.75)+(−1)+0.5) =1(1.5+0−1−1.75−1+0.5) =1(2−3.75) =1(−1.75) =−1.75

Hence,

The value of midpoints is −1.75 .

Chapter 5.2, Problem 5E

Chapter 5.2, Problem 7E

Chapter 5 Solutions

Single Variable Calculus: Concepts and Contexts, Enhanced Edition

Ch. 5.1 - Prob. 1E Ch. 5.1 - (a) Use six rectangles to find estimates of each...Ch. 5.1 - (a) Estimate the area under the graph of f(x) =...Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - The speed of a runner increased steadily during...Ch. 5.1 - Prob. 12E

Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - The velocity graph of a car accelerating from rest...Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 28E Ch. 5.2 - Evaluate the Riemann sum for f(x) = x 1, 6 x ...Ch. 5.2 - Prob. 2E Ch. 5.2 - Prob. 3E Ch. 5.2 - (a) Find the Riemann sum for f(x) = 1/x, 1 x 2,...Ch. 5.2 - Prob. 5E Ch. 5.2 - Prob. 6E Ch. 5.2 - A table of values of an increasing function f is...Ch. 5.2 - Prob. 8E Ch. 5.2 - Use the Midpoint Rule with the given value of n to...Ch. 5.2 - Use the Midpoint Rule with the given value of n to...Ch. 5.2 - Use the Midpoint Rule with the given value of n to...Ch. 5.2 - Use the Midpoint Rule with the given value of n to...Ch. 5.2 - With a programmable calculator or computer (see...Ch. 5.2 - Prob. 15E Ch. 5.2 - Use a calculator or computer to make a table of...Ch. 5.2 - Prob. 17E Ch. 5.2 - Prob. 18E Ch. 5.2 - Prob. 19E Ch. 5.2 - Prob. 20E Ch. 5.2 - Prob. 21E Ch. 5.2 - Prob. 22E Ch. 5.2 - Prob. 23E Ch. 5.2 - Prob. 24E Ch. 5.2 - Prob. 25E Ch. 5.2 - Prob. 26E Ch. 5.2 - Prob. 27E Ch. 5.2 - Prob. 28E Ch. 5.2 - Prob. 31E Ch. 5.2 - The graph of g consists of two straight lines and...Ch. 5.2 - Prob. 33E Ch. 5.2 - Prob. 34E Ch. 5.2 - Prob. 35E Ch. 5.2 - Prob. 36E Ch. 5.2 - Prob. 37E Ch. 5.2 - Prob. 38E Ch. 5.2 - Prob. 39E Ch. 5.2 - Prob. 40E Ch. 5.2 - Prob. 41E Ch. 5.2 - Prob. 42E Ch. 5.2 - Prob. 43E Ch. 5.2 - Prob. 44E Ch. 5.2 - Prob. 45E Ch. 5.2 - Prob. 46E Ch. 5.2 - Prob. 47E Ch. 5.2 - If , F(x)=2xf(t)dt, where f is the function whose...Ch. 5.2 - Each of the regions A, B, and C bounded by the...Ch. 5.2 - Prob. 50E Ch. 5.2 - Prob. 51E Ch. 5.2 - Prob. 52E Ch. 5.2 - Prob. 53E Ch. 5.2 - Prob. 54E Ch. 5.2 - Prob. 55E Ch. 5.2 - Prob. 56E Ch. 5.3 - Prob. 1E Ch. 5.3 - Prob. 2E Ch. 5.3 - Prob. 3E Ch. 5.3 - Prob. 4E Ch. 5.3 - Prob. 5E Ch. 5.3 - Prob. 6E Ch. 5.3 - Prob. 7E Ch. 5.3 - Prob. 8E Ch. 5.3 - Prob. 9E Ch. 5.3 - Prob. 10E Ch. 5.3 - Prob. 11E Ch. 5.3 - Prob. 12E Ch. 5.3 - Prob. 13E Ch. 5.3 - Prob. 14E Ch. 5.3 - Prob. 15E Ch. 5.3 - Prob. 16E Ch. 5.3 - Prob. 17E Ch. 5.3 - Prob. 18E Ch. 5.3 - Prob. 19E Ch. 5.3 - Prob. 20E Ch. 5.3 - Prob. 21E Ch. 5.3 - Prob. 22E Ch. 5.3 - Prob. 23E Ch. 5.3 - Prob. 24E Ch. 5.3 - Prob. 25E Ch. 5.3 - Prob. 26E Ch. 5.3 - Prob. 27E Ch. 5.3 - Prob. 28E Ch. 5.3 - Prob. 29E Ch. 5.3 - Prob. 30E Ch. 5.3 - Prob. 31E Ch. 5.3 - Prob. 32E Ch. 5.3 - Prob. 33E Ch. 5.3 - Prob. 34E Ch. 5.3 - Prob. 35E Ch. 5.3 - Prob. 36E Ch. 5.3 - Prob. 37E Ch. 5.3 - Prob. 38E Ch. 5.3 - Prob. 39E Ch. 5.3 - Prob. 40E Ch. 5.3 - Prob. 41E Ch. 5.3 - Prob. 42E Ch. 5.3 - Prob. 43E Ch. 5.3 - Prob. 44E Ch. 5.3 - Prob. 45E Ch. 5.3 - Prob. 46E Ch. 5.3 - Prob. 47E Ch. 5.3 - Prob. 48E Ch. 5.3 - Prob. 49E Ch. 5.3 - Prob. 50E Ch. 5.3 - Prob. 51E Ch. 5.3 - Prob. 52E Ch. 5.3 - Prob. 53E Ch. 5.3 - Prob. 54E Ch. 5.3 - Prob. 55E Ch. 5.3 - Prob. 56E Ch. 5.3 - Prob. 57E Ch. 5.3 - Prob. 58E Ch. 5.3 - Prob. 59E Ch. 5.3 - Prob. 60E Ch. 5.3 - Prob. 61E Ch. 5.3 - Prob. 62E Ch. 5.3 - Prob. 63E Ch. 5.3 - Prob. 64E Ch. 5.3 - Prob. 65E Ch. 5.3 - Prob. 66E Ch. 5.3 - Prob. 67E Ch. 5.3 - Prob. 68E Ch. 5.3 - Prob. 69E Ch. 5.3 - Prob. 70E Ch. 5.3 - Prob. 71E Ch. 5.3 - Prob. 72E Ch. 5.3 - Prob. 73E Ch. 5.3 - Prob. 74E Ch. 5.3 - Prob. 75E Ch. 5.3 - Prob. 76E Ch. 5.4 - Explain exactly what is meant by the statement...Ch. 5.4 - Let g(x)=0xf(t)dt, where f is the function whose...Ch. 5.4 - Prob. 3E Ch. 5.4 - Prob. 4E Ch. 5.4 - Prob. 5E Ch. 5.4 - Sketch the area represented by g(x). 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Publisher:PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Precalculus
Calculus
ISBN:9780135189405
Author:Michael Sullivan
Publisher:PEARSON
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning