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Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

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Chapter
Section
BuyFindarrow_forward

Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 14, Problem 3PS
Textbook Problem
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Using a Change of variables The figure shows the region R bounded by the curves

y = x , y = 2 x , y = x 2 3 , and y = x 2 4 .

Use the change of variables x = u 1 / 3 v 2 / 3 and y = u 2 / 3 v 1 / 3 to find the area of the region R .

Chapter 14, Problem 3PS, Using a Change of variables The figure shows the region R bounded by the curves

To determine

To calculate: Find the area of the given region ‘R’.

Explanation of Solution

Given:

The region R is shown below

As we know the region R is bounded by the curves

y=x, y=2x, y=x33 and y=x24

We need to use the change of variables x=u1/3v2/3 and y=u2/3v1/3

Formula used:

If x=g(u,v) and y=h(u,v), then the Jacobian of x and y with respect to u and v, denoted by (x,y)(u,v) is

(x,y)(u,v)=|xuxvyuyv|=xuyvyuxv

Calculation:

As we know, given that

x=u1/3v2/3 and y=u2/3v1/3

Therefore further we have,

T(u,v)=(x,y)=(u1/3v2/3,u2/3v1/3)

Now,

All the four boundaries of R in xy plane give rise the four boundaries of S in uv plane.

Hence we have,

[Boundary in xy planeBoundary in uv planey=xu=1y=2xu=2y=x23v=3y=x24v=4]

From the above we get region S in uv plane as depicted below

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Chapter 14 Solutions

Multivariable Calculus
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Consider the surface...Ch. 14.5 - Surface AreaAnswer each question about the surface...Ch. 14.5 - Surface Area Consider the surface f(x,y)=x+y. What...Ch. 14.5 - Product DesignA company produces a spherical...Ch. 14.5 - Modeling Data A company builds a ware house with...Ch. 14.5 - Surface Area Find the surface area of the solid of...Ch. 14.5 - Surface AreaShow that the surface area of the cone...Ch. 14.6 - CONCEPT CHECK Triple Integrals What does Q=QdV...Ch. 14.6 - Changing the Order of Integration Why is it...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Evaluating a Triple Iterated Integral Using...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Setting Up a Triple IntegralIn Exercises 13-18,...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Volume In Exercises 19-24, use a triple integral...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Changing the Order of Integration In Exercises...Ch. 14.6 - Changing the Order of integration In Exercises...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 31-34, write a...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Orders of Integration In Exercises 35 and 36, the...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 37-40, find the mass...Ch. 14.6 - Center of Mass In Exercises 41 and 42, set up the...Ch. 14.6 - Center of Mass In Exercises 41 and 42, set up the...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Think About It The center of mass of a solid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Centroid In Exercises 47-52, find the centroid of...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 59 and 60, set up...Ch. 14.6 - Moments of Inertia In Exercises 59 and 60, set up...Ch. 14.6 - Setting Up Triple IntegralsIn Exercises 61 and 62,...Ch. 14.6 - Setting Up Triple IntegralsIn Exercises 61 and 62,...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - Average Value In Exercises 63-66, find the average...Ch. 14.6 - EXPLORING CONCEPTS Moment of Inertia Determine...Ch. 14.6 - Using Different Methods Find the volume of the...Ch. 14.6 - EXPLORING CONCEPTS (continued) Think About It...Ch. 14.6 - HOW DO YOU SEE IT? Consider two solids of equal...Ch. 14.6 - Maximizing a Triple Integral Find the solid region...Ch. 14.6 - Find a Value Solve for a in the triple integral....Ch. 14.6 - PUTNAM EXAM CHALLENGE Evaluate limn0101...01cos2{...Ch. 14.7 - CONCEPT CHECK Volume Explain why triple integrals...Ch. 14.7 - CONCEPT CHECK Differential of Volume What is the...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14.7 - Evaluating a Triple Iterated Integral Using...Ch. 14.7 - Evaluating a Triple Iterated Integral Using...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - VolumeIn Exercises 1114, sketch the solid region...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - Volume In Exercises 11-14, sketch the solid region...Ch. 14.7 - VolumeIn Exercises 1520, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - VolumeIn Exercises 1520, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - VolumeIn Exercises 1520, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Mass In Exercises 21 and 22, use cylindrical...Ch. 14.7 - Mass In Exercises 21 and 22, use cylindrical...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Moment of Inertia In Exercises 29 and 30, use...Ch. 14.7 - VolumeIn Exercises 3134, use spherical coordinates...Ch. 14.7 - VolumeIn Exercises 3134, use spherical coordinates...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - VolumeIn Exercises 3134, use spherical coordinates...Ch. 14.7 - MassIn Exercises 35 and 36, use spherical...Ch. 14.7 - MassIn Exercises 35 and 36, use spherical...Ch. 14.7 - Center of Mass In Exercises 37 and 38, use...Ch. 14.7 - Center of MassIn Exercises 37 and 38, use...Ch. 14.7 - Moment of InertiaIn Exercises 39 and 40, use...Ch. 14.7 - Moment of InertiaIn Exercises 39 and 40, use...Ch. 14.7 - Converting CoordinatesIn Exercises 4144, convert...Ch. 14.7 - Converting CoordinatesIn Exercises 4144, convert...Ch. 14.7 - Converting CoordinatesIn Exercises 4144, convert...Ch. 14.7 - EXPLORING CONCEPTS Using CoordinatesDescribe the...Ch. 14.7 - HOW DO YOU SEE IT? The solid is bounded below by...Ch. 14.7 - PUTNAM EXAM CHALLENGE Find the volume of the...Ch. 14.8 - CONCEPT CHECK JacobianDescribe how to find the...Ch. 14.8 - CONCEPT CHECK Change of VariableWhen is it...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Finding a Jacobian In Exercises 3-10, find the...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Using a Transformation In Exercises 11-14, sketch...Ch. 14.8 - Verifying a Change of VariablesIn Exercises 15 and...Ch. 14.8 - Verifying a Change of VariablesIn Exercises 15 and...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Evaluating a Double Integral Using a Change of...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - Finding Volume Using a Change of Variables In...Ch. 14.8 - EXPLORING CONCEPTS Using a Transformation The...Ch. 14.8 - HOW DO YOU SEE IT? The region R is transformed...Ch. 14.8 - Using an EllipseConsider the region R in the...Ch. 14.8 - VolumeUse the result of Exercise 33 to find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, find the...Ch. 14.8 - PUTNAM EXAM CHALLENGE Let A be the area of the...Ch. 14 - Evaluating an Integral In Exercises 1 and 2,...Ch. 14 - Evaluating an Integral In Exercises 1 and 2,...Ch. 14 - Evaluating an Integral In Exercises 3 - 6,...Ch. 14 - Evaluating an Integral In Exercises 3 - 6,...Ch. 14 - Evaluating an Iterated Integral In Exercises 3-6,...Ch. 14 - Evaluating an Iterated Integral In Exercises 3-6,...Ch. 14 - Finding the Area of a RegionIn Exercises 710, use...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...Ch. 14 - Finding the Area of a RegionIn Exercises 710, use...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Changing the Order of Integration In Exercises...Ch. 14 - Evaluating a Double Integral In Exercises 15 and...Ch. 14 - Evaluating a Double Integral In Exercises 15 and...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Finding VolumeIn Exercises 1720, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Average ValueIn Exercises 21 and 22, find the...Ch. 14 - Average ValueIn Exercises 21 and 22, find the...Ch. 14 - Average Temperature The temperature in degrees...Ch. 14 - Average Profit A firm's profit P (in dollars) from...Ch. 14 - Converting to Polar CoordinatesIn Exercises 25 and...Ch. 14 - Converting to Polar CoordinatesIn Exercises 25 and...Ch. 14 - VolumeIn Exercises 27 and 28, use a double...Ch. 14 - Volume In Exercises 27 and 28, use a double...Ch. 14 - AreaIn Exercises 29 and 30, use a double integral...Ch. 14 - AreaIn Exercises 29 and 30, use a double integral...Ch. 14 - AreaIn Exercises 31 and 32, sketch a graph of the...Ch. 14 - AreaIn Exercises 31 and 32, sketch a graph of the...Ch. 14 - Area and VolumeConsider the region R in the xy...Ch. 14 - Converting to Polar Coordinates Write the sum of...Ch. 14 - Finding the Mass of a LaminaIn Exercises 35 and...Ch. 14 - Finding the Mass of a LaminaIn Exercises 35 and...Ch. 14 - Finding the Center of MassIn Exercises 3740, find...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding the Center of Mass In Exercises 37-40,...Ch. 14 - Finding Moments of Inertia and Radii of Gyration...Ch. 14 - Finding Moments of Inertia and Radii of GyrationIn...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Finding Surface AreaIn Exercises 4346, find the...Ch. 14 - Building DesignA new auditorium is built with a...Ch. 14 - Surface Area The roof over the stage of an open...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - VolumeIn Exercises 55 and 56, use a triple...Ch. 14 - VolumeIn Exercises 55 and 56, use a triple...Ch. 14 - Changing the Order of Integration In Exercises 57...Ch. 14 - Center of Mass In Exercises 59 and 60, find the...Ch. 14 - Center of Mass In Exercises 59 and 60, find the...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated IntegralIn Exercises...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - VolumeIn Exercises 67 and 68, use cylindrical...Ch. 14 - VolumeIn Exercises 67 and 68, use cylindrical...Ch. 14 - VolumeIn Exercises 69 and 70, use spherical...Ch. 14 - VolumeIn Exercises 69 and 70, use spherical...Ch. 14 - Finding a JcobianIn Exercises 7174, find the...Ch. 14 - Finding a JcobianIn Exercises 7174, find the...Ch. 14 - Finding a JcobianIn Exercises 7174, find the...Ch. 14 - Finding a JcobianIn Exercises 7174, find the...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - Evaluating a Double Integral Using a Change of...Ch. 14 - VolumeFind the volume of the solid of intersection...Ch. 14 - Surface AreaLet a,b,c, and d be positive real...Ch. 14 - Using a Change of variables The figure shows the...Ch. 14 - ProofProve that limn0101xnyndxdy=0.Ch. 14 - Deriving a Sum Derive Eulers famous result that...Ch. 14 - Evaluating a Double IntegralEvaluate the integral...Ch. 14 - Evaluating Double IntegralsEvaluate the integrals...Ch. 14 - VolumeShow that the volume of a spherical block...Ch. 14 - Evaluating an IntegralIn Exercises 9 and 10,...Ch. 14 - Evaluating an Integral In Exercises 9 and 10,...Ch. 14 - Joint Density Function Consider the function...Ch. 14 - Volume Find the volume of the solid generated by...Ch. 14 - Proof The angle between a plane P and the xy...Ch. 14 - Surface Area Use the result of Exercise 14 to...Ch. 14 - SprinklerConsider a circular lawn with a radius of...Ch. 14 - Volume The figure shows a solid bounded below by...

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