BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area.

y = 2 x + 1 , 0 x 1

To determine

To estimate:

Area of the region that lies under the given curve using graph and then find the exact area

Explanation

1) Concept:

i) The substitution rule:

 If u=g(x) is a differentiable function whose range is I and f is continuous on I, then f(gx)g'xdx=f(u)du. Here,g(x) is substituted as u and then the differentiation is g(x)dx =du

ii) Indefinite integral

xn dx=xn+1n+1+C   (n-1)

2) Given:

y=2x+1 , 0x1

3) Graph:

4) Calculation:

From the graph, the area of the shaded region is split into two parts. One is a square with area 1 and the other is approximately a triangle with area less than  0.4

Therefore, the area of the shaded region from the graph is approximately 1.4.

Now, to find exact area, integrate given function.

A=012x+1dx

Here, use the substitution method.

Substitute 2x+1=u.

Differentiating with respect to x

2dx=du

dx=du2

The limits change and the new limits of integration are calculated by substituting

for x=0, u=20+1=1 and for x=1, u=21+1=3

Therefore, the given integral becomes

=13udu2

Factor out 12

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started
Sect-4.1 P-11ESect-4.1 P-12ESect-4.1 P-13ESect-4.1 P-14ESect-4.1 P-15ESect-4.1 P-16ESect-4.1 P-17ESect-4.1 P-18ESect-4.1 P-19ESect-4.1 P-20ESect-4.1 P-21ESect-4.1 P-22ESect-4.1 P-23ESect-4.1 P-24ESect-4.1 P-25ESect-4.1 P-26ESect-4.1 P-27ESect-4.1 P-28ESect-4.1 P-29ESect-4.1 P-30ESect-4.1 P-31ESect-4.1 P-32ESect-4.2 P-1ESect-4.2 P-2ESect-4.2 P-3ESect-4.2 P-4ESect-4.2 P-5ESect-4.2 P-6ESect-4.2 P-7ESect-4.2 P-8ESect-4.2 P-9ESect-4.2 P-10ESect-4.2 P-11ESect-4.2 P-12ESect-4.2 P-13ESect-4.2 P-14ESect-4.2 P-15ESect-4.2 P-16ESect-4.2 P-17ESect-4.2 P-18ESect-4.2 P-19ESect-4.2 P-20ESect-4.2 P-21ESect-4.2 P-22ESect-4.2 P-23ESect-4.2 P-24ESect-4.2 P-25ESect-4.2 P-26ESect-4.2 P-27ESect-4.2 P-28ESect-4.2 P-29ESect-4.2 P-30ESect-4.2 P-31ESect-4.2 P-32ESect-4.2 P-33ESect-4.2 P-34ESect-4.2 P-35ESect-4.2 P-36ESect-4.2 P-37ESect-4.2 P-38ESect-4.2 P-39ESect-4.2 P-40ESect-4.2 P-41ESect-4.2 P-42ESect-4.2 P-43ESect-4.2 P-44ESect-4.2 P-45ESect-4.2 P-46ESect-4.2 P-47ESect-4.2 P-48ESect-4.2 P-49ESect-4.2 P-50ESect-4.2 P-51ESect-4.2 P-52ESect-4.2 P-53ESect-4.2 P-54ESect-4.2 P-55ESect-4.2 P-56ESect-4.2 P-57ESect-4.2 P-58ESect-4.2 P-59ESect-4.2 P-60ESect-4.2 P-61ESect-4.2 P-62ESect-4.2 P-63ESect-4.2 P-64ESect-4.2 P-65ESect-4.2 P-66ESect-4.2 P-67ESect-4.2 P-68ESect-4.2 P-69ESect-4.2 P-70ESect-4.2 P-71ESect-4.2 P-72ESect-4.2 P-73ESect-4.2 P-74ESect-4.2 P-75ESect-4.3 P-1ESect-4.3 P-2ESect-4.3 P-3ESect-4.3 P-4ESect-4.3 P-5ESect-4.3 P-6ESect-4.3 P-7ESect-4.3 P-8ESect-4.3 P-9ESect-4.3 P-10ESect-4.3 P-11ESect-4.3 P-12ESect-4.3 P-13ESect-4.3 P-14ESect-4.3 P-15ESect-4.3 P-16ESect-4.3 P-17ESect-4.3 P-18ESect-4.3 P-19ESect-4.3 P-20ESect-4.3 P-21ESect-4.3 P-22ESect-4.3 P-23ESect-4.3 P-24ESect-4.3 P-25ESect-4.3 P-26ESect-4.3 P-27ESect-4.3 P-28ESect-4.3 P-29ESect-4.3 P-30ESect-4.3 P-31ESect-4.3 P-32ESect-4.3 P-33ESect-4.3 P-34ESect-4.3 P-35ESect-4.3 P-36ESect-4.3 P-37ESect-4.3 P-38ESect-4.3 P-39ESect-4.3 P-40ESect-4.3 P-41ESect-4.3 P-42ESect-4.3 P-43ESect-4.3 P-44ESect-4.3 P-45ESect-4.3 P-46ESect-4.3 P-47ESect-4.3 P-48ESect-4.3 P-49ESect-4.3 P-50ESect-4.3 P-51ESect-4.3 P-52ESect-4.3 P-53ESect-4.3 P-54ESect-4.3 P-55ESect-4.3 P-56ESect-4.3 P-57ESect-4.3 P-58ESect-4.3 P-59ESect-4.3 P-60ESect-4.3 P-61ESect-4.3 P-62ESect-4.3 P-63ESect-4.3 P-64ESect-4.3 P-65ESect-4.3 P-66ESect-4.3 P-67ESect-4.3 P-68ESect-4.3 P-69ESect-4.3 P-70ESect-4.3 P-71ESect-4.3 P-72ESect-4.3 P-73ESect-4.3 P-74ESect-4.3 P-75ESect-4.3 P-76ESect-4.3 P-77ESect-4.3 P-78ESect-4.3 P-79ESect-4.3 P-80ESect-4.3 P-81ESect-4.3 P-82ESect-4.3 P-83ESect-4.3 P-84ESect-4.4 P-1ESect-4.4 P-2ESect-4.4 P-3ESect-4.4 P-4ESect-4.4 P-5ESect-4.4 P-6ESect-4.4 P-7ESect-4.4 P-8ESect-4.4 P-9ESect-4.4 P-10ESect-4.4 P-11ESect-4.4 P-12ESect-4.4 P-13ESect-4.4 P-14ESect-4.4 P-15ESect-4.4 P-16ESect-4.4 P-17ESect-4.4 P-18ESect-4.4 P-19ESect-4.4 P-20ESect-4.4 P-21ESect-4.4 P-22ESect-4.4 P-23ESect-4.4 P-24ESect-4.4 P-25ESect-4.4 P-26ESect-4.4 P-27ESect-4.4 P-28ESect-4.4 P-29ESect-4.4 P-30ESect-4.4 P-31ESect-4.4 P-32ESect-4.4 P-33ESect-4.4 P-34ESect-4.4 P-35ESect-4.4 P-36ESect-4.4 P-37ESect-4.4 P-38ESect-4.4 P-39ESect-4.4 P-40ESect-4.4 P-41ESect-4.4 P-42ESect-4.4 P-43ESect-4.4 P-44ESect-4.4 P-45ESect-4.4 P-46ESect-4.4 P-47ESect-4.4 P-48ESect-4.4 P-49ESect-4.4 P-50ESect-4.4 P-51ESect-4.4 P-52ESect-4.4 P-53ESect-4.4 P-54ESect-4.4 P-55ESect-4.4 P-56ESect-4.4 P-57ESect-4.4 P-58ESect-4.4 P-59ESect-4.4 P-60ESect-4.4 P-61ESect-4.4 P-62ESect-4.4 P-63ESect-4.4 P-64ESect-4.4 P-65ESect-4.4 P-66ESect-4.4 P-67ESect-4.4 P-68ESect-4.4 P-69ESect-4.4 P-70ESect-4.4 P-71ESect-4.4 P-72ESect-4.4 P-73ESect-4.4 P-74ESect-4.5 P-1ESect-4.5 P-2ESect-4.5 P-3ESect-4.5 P-4ESect-4.5 P-5ESect-4.5 P-6ESect-4.5 P-7ESect-4.5 P-8ESect-4.5 P-9ESect-4.5 P-10ESect-4.5 P-11ESect-4.5 P-12ESect-4.5 P-13ESect-4.5 P-14ESect-4.5 P-15ESect-4.5 P-16ESect-4.5 P-17ESect-4.5 P-18ESect-4.5 P-19ESect-4.5 P-20ESect-4.5 P-21ESect-4.5 P-22ESect-4.5 P-23ESect-4.5 P-24ESect-4.5 P-25ESect-4.5 P-26ESect-4.5 P-27ESect-4.5 P-28ESect-4.5 P-29ESect-4.5 P-30ESect-4.5 P-31ESect-4.5 P-32ESect-4.5 P-33ESect-4.5 P-34ESect-4.5 P-35ESect-4.5 P-36ESect-4.5 P-37ESect-4.5 P-38ESect-4.5 P-39ESect-4.5 P-40ESect-4.5 P-41ESect-4.5 P-42ESect-4.5 P-43ESect-4.5 P-44ESect-4.5 P-45ESect-4.5 P-46ESect-4.5 P-47ESect-4.5 P-48ESect-4.5 P-49ESect-4.5 P-50ESect-4.5 P-51ESect-4.5 P-52ESect-4.5 P-53ESect-4.5 P-54ESect-4.5 P-55ESect-4.5 P-56ESect-4.5 P-57ESect-4.5 P-58ESect-4.5 P-59ESect-4.5 P-60ESect-4.5 P-61ESect-4.5 P-62ESect-4.5 P-63ESect-4.5 P-64ESect-4.5 P-65ESect-4.5 P-66ESect-4.5 P-67ESect-4.5 P-68ESect-4.5 P-69ESect-4.5 P-70ESect-4.5 P-71ESect-4.5 P-72ESect-4.5 P-73ESect-4.5 P-74ESect-4.5 P-75ESect-4.5 P-76ESect-4.5 P-77ESect-4.5 P-78ESect-4.5 P-79ESect-4.5 P-80ESect-4.5 P-81ESect-4.5 P-82ESect-4.5 P-83ESect-4.5 P-84ESect-4.5 P-85ESect-4.R P-1CCSect-4.R P-2CCSect-4.R P-3CCSect-4.R P-4CCSect-4.R P-5CCSect-4.R P-6CCSect-4.R P-7CCSect-4.R P-8CCSect-4.R P-9CCSect-4.R P-1TFQSect-4.R P-2TFQSect-4.R P-3TFQSect-4.R P-4TFQSect-4.R P-5TFQSect-4.R P-6TFQSect-4.R P-7TFQSect-4.R P-8TFQSect-4.R P-9TFQSect-4.R P-10TFQSect-4.R P-11TFQSect-4.R P-12TFQSect-4.R P-13TFQSect-4.R P-14TFQSect-4.R P-15TFQSect-4.R P-16TFQSect-4.R P-17TFQSect-4.R P-18TFQSect-4.R P-1ESect-4.R P-2ESect-4.R P-3ESect-4.R P-4ESect-4.R P-5ESect-4.R P-6ESect-4.R P-7ESect-4.R P-8ESect-4.R P-9ESect-4.R P-10ESect-4.R P-11ESect-4.R P-12ESect-4.R P-13ESect-4.R P-14ESect-4.R P-15ESect-4.R P-16ESect-4.R P-17ESect-4.R P-18ESect-4.R P-19ESect-4.R P-20ESect-4.R P-21ESect-4.R P-22ESect-4.R P-23ESect-4.R P-24ESect-4.R P-25ESect-4.R P-26ESect-4.R P-27ESect-4.R P-28ESect-4.R P-29ESect-4.R P-30ESect-4.R P-31ESect-4.R P-32ESect-4.R P-33ESect-4.R P-34ESect-4.R P-35ESect-4.R P-36ESect-4.R P-37ESect-4.R P-38ESect-4.R P-39ESect-4.R P-40ESect-4.R P-41ESect-4.R P-42ESect-4.R P-43ESect-4.R P-44ESect-4.R P-45ESect-4.R P-46ESect-4.R P-47ESect-4.R P-48ESect-4.R P-49ESect-4.R P-50ESect-4.R P-51ESect-4.R P-52ESect-4.R P-53ESect-4.R P-54ESect-4.R P-55ESect-4.R P-56ESect-4.R P-57ESect-4.R P-58ESect-4.P P-1PSect-4.P P-2PSect-4.P P-3PSect-4.P P-4PSect-4.P P-5PSect-4.P P-6PSect-4.P P-7PSect-4.P P-8PSect-4.P P-9PSect-4.P P-10PSect-4.P P-11PSect-4.P P-12PSect-4.P P-13PSect-4.P P-14PSect-4.P P-15PSect-4.P P-16PSect-4.P P-17PSect-4.P P-18P

Additional Math Solutions

Find more solutions based on key concepts

Show solutions add

In Exercises 7-12, solve for y in terms of x. x=2y14

Calculus: An Applied Approach (MindTap Course List)

What are the three ways to solve a quadratic equation?

Precalculus: Mathematics for Calculus (Standalone Book)

If the line passing through the points (a, 1) and (5, 8) is parallel to the line passing through the points (4,...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Evaluate each expression: 5((4+6)+2(52))

Elementary Technical Mathematics

Differentiate. f() = cos sin

Single Variable Calculus: Early Transcendentals

Find the sum of even integers between 21 and 99.

Finite Mathematics for the Managerial, Life, and Social Sciences

The tangential component of acceleration for at t = 0 is: 0

Study Guide for Stewart's Multivariable Calculus, 8th

Find the sum of the areas of approximating rectangles for the area under f (x) = 48 − x2, between x = 1 and x =...

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Outline the major advantages and disadvantages of administering a survey by mail.

Research Methods for the Behavioral Sciences (MindTap Course List)

Solve each system by elimination. {8x+12y=242x+3y=4

College Algebra (MindTap Course List)