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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.8, Problem 39E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Changing the Order of Integration In Exercises 37-44, sketch the region R whose area is given by the double integral. Then change the order of integration and show that both orders yield the same value. See Example 5.

∫ 0 1 ∫ 2 y 2 d y   d x

To determine

To graph: The region whose area is given by double integration ∫01∫2y2dxdy, the change the order of integration and shows both orders yield same value.

Explanation

Given Information:

The provided double integration is ∫01∫2y2dxdy.

Graph:

Consider the double integration,

∫01∫2y2dxdy.

From limits of integration, the bounds for x are 0≤x≤1 and bounds for y are 2y≤x≤y.

The table shown the coordinate of y=3x,

x -Coordinates y - Coordinates (x,y) Coordinates
2 1 (2,1)
0 0 (0,0)

The graph of region bounded by 0≤x≤1 and 2y≤x≤y is shown in below,

The area for the region 0≤x≤1 and 2y≤x≤y is

∫01∫2y2dxdy

Evaluate the above integration integrate with respect to x by holding y constant,

∫01∫2y2dxdy=∫01[x]2y2dy

Now, replace the x by limit of integration,

∫01[x]2y2dy=∫01[2−2y]dy

Evaluate the above integration integrate with respect to y by holding x constant,

∫01[2−2y]dy=[2y−y2]01

Now, replace the y by limit of integration,

[2y−y2]01=(2−1)=1

Now change the order of integration dxdy to dydx

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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Statistics for The Behavioral Sciences (MindTap Course List)

Agriculture: Apple Trees The following data represent trunk circumferences (in mm) for a random sample of 60 fo...

Understanding Basic Statistics

Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and ...

Single Variable Calculus

For Problems 1-50, solve each equation. Objective 1 3x5(2x+1)=13

Intermediate Algebra

In Exercises I to 14, use the roster method to write each of the given sets. For some exercises you may need to...

Mathematical Excursions (MindTap Course List)

Simplify each expression in Exercises 1730, expressing your answer in positive exponent form. x3x2

Finite Mathematics

In a famous experiment carried out in 1882, Michelson and Newcomb obtained 66 observations on the time it took ...

Probability and Statistics for Engineering and the Sciences

Factor each expression in Problems 7-18 as a product of binomials. 16.

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate the integral. 19xdx

Single Variable Calculus: Early Transcendentals

Using Table 11-1, calculate the compound amount and compound interest for the following investment. Principal T...

Contemporary Mathematics for Business & Consumers

Label each of the following statements as either true or false. Any group of order 4 must be cyclic.

Elements Of Modern Algebra

By definition, a hyperbola is the locus of points whose positive difference of distances from two fixed points ...

Elementary Geometry For College Students, 7e

Change 1.5 m3 to cm3.

Elementary Technical Mathematics

For Questions 1 through 4, fill in the blank with an appropriate word, number, or expression. The 5th roots of ...

Trigonometry (MindTap Course List)

Finding the Area of a Polar Region In Exercises 19-26, use a graphing utility to graph the polar equation. Find...

Calculus: Early Transcendental Functions (MindTap Course List)

Differential Equation In Exercises 35-38, find the general solution of the differential equation. yt=arctanx2

Calculus of a Single Variable

In Exercises 724, compute the indicated products. [1.20.30.40.5][0.20.60.40.5]

Finite Mathematics for the Managerial, Life, and Social Sciences

A triangle has angles measuring 30, 30, and 120. If the congruent sides measure 6 units each, find the length o...

Elementary Geometry for College Students

Find a definite integral for the consumer surplus if 180 units are available and the demand function is p(x) = ...

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

Evaluating a Line Integral in Differential Form In Exercises 5764, evaluate C(2xy)dx+(x+3y)dy. C: arc on y=1x2 ...

Calculus (MindTap Course List)

The surface given by z = −r2 in cylindrical coordinates is a(n): elliptic cone hyperboloid of one sheet hyperbo...

Study Guide for Stewart's Multivariable Calculus, 8th

Finding Unit Vectors In Exercises 79-82, find a unit vector (a) in the direction or r and (b) in the direction ...

Calculus

EXPLORING CONCEPTS Think About ItIn Exercises 57 and 58, consider two forces of equal magnitude acting on a poi...

Multivariable Calculus

Exercise Use the graph function fshown to determine each of the following. f2

College Algebra (MindTap Course List)

In addition to the key words, you should also be able to define each of the following terms: frequency distribu...

Research Methods for the Behavioral Sciences (MindTap Course List)

A simple random sample of 50 items from a population with = 6 resulted in a sample mean of 32. a. Provide a 90...

Statistics for Business & Economics, Revised (MindTap Course List)

The Franklins bought a house for 265,000. They put 20 down and financed the rest with a thirty-year loan at 538...

Mathematics: A Practical Odyssey

The USA Today reports that the average expenditure on Valentines Day is 100.89. Do male and female consumers di...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

The article Brain Shunt Tested to Treat Alzheimers (San Francisco Chronicle, October 23, 2002) summarizes the f...

Introduction To Statistics And Data Analysis

Briefly explain how a participant characteristic, such as personality, could be a confounding variable in a bet...

Research Methods for the Behavioral Sciences (MindTap Course List)

The following Exercises 7 through 19 involve rectangles, parallelograms, trapezoids, and triangles. Given certa...

Mathematics For Machine Technology

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