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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 14.1, Problem 79E

Textbook Problem

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True or False? In Exercises 79 and 80, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

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To determine

Whether the statement “∫ a b ∫ c d f ( x , y ) d y d x = ∫ c d ∫ a b f ( x , y ) d x d y ” is true or false.

Let

Consider the left-side of the provided statement,

Then,

Consider the right-side of the provided statement,

Calculus: Early Transcendental Functions

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Consider the surface...Ch. 14.5 - Surface Area Answer each question about the...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 warehouse with...Ch. 14.5 - Surface Area Find the surface area of the solid of...Ch. 14.5 - Surface Area Show that the surface area of the...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 MassIn Exercises 41 and 42, set up the...Ch. 14.6 - Center of MassIn 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 ItThe center of mass of a solid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - CentroidIn Exercises 47-52, find the centroid of...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find Ix,...Ch. 14.6 - Moments of Inertia In Exercises 53- 56, find...Ch. 14.6 - Moments of InertiaIn Exercises 53- 56, find Ix,Iy,...Ch. 14.6 - Moments of InertiaIn Exercises 53- 56, find Ix,Iy,...Ch. 14.6 - Moments of InertiaIn Exercises 57 and 58, verify...Ch. 14.6 - Moments of Inertia In Exercises 57 and 58, verify...Ch. 14.6 - Moments of InertiaIn Exercises 59 and 60, set up a...Ch. 14.6 - Moments of InertiaIn Exercises 59 and 60, set up a...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 ValueIn Exercises 63-66, find the average...Ch. 14.6 - Average ValueIn Exercises 63-66, find the average...Ch. 14.6 - Average ValueIn 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 - Finding a Value Solve for a in the triple integral...Ch. 14.6 - PUTNAM EXAM CHALLENGE Evaluate...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 IntegralIn 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 IntegralIn 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 - Volume In Exercises 11-14, 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 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, use cylindrical...Ch. 14.7 - Volume In Exercises 15-20, 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 CoordinatesIn Exercises 23-28,...Ch. 14.7 - Using Cylindrical Coordinates In Exercises 23-28,...Ch. 14.7 - Using Cylindrical CoordinatesIn Exercises 23-28,...Ch. 14.7 - Moment of Inertia In Exercises 29 and 30, use...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Volume In Exercises 31-34, use spherical...Ch. 14.7 - Mass In Exercises 35 and 36, use spherical...Ch. 14.7 - Mass In 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 Inertia In Exercises 39 and 40, use...Ch. 14.7 - Moment of Inertia In Exercises 39 and 40, use...Ch. 14.7 - Converting Coordinates In Exercises 41-44, convert...Ch. 14.7 - Converting Coordinates In Exercises 41-44, convert...Ch. 14.7 - Converting Coordinates In Exercises 41-44, convert...Ch. 14.7 - EXPLORING CONCEPTS Using Coordinates Describe 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 Jacobian Describe how to find the...Ch. 14.8 - CONCEPT CHECK Change of Variable When 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...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 JacobianIn 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 TransformationIn Exercises 11-14, sketch...Ch. 14.8 - Verifying a Change of Variables In Exercises 15...Ch. 14.8 - Verifying a Change of Variables In Exercises 15...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 VolumeUsing 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 Ellipse Consider the region R in the...Ch. 14.8 - Volume Use the result of Exercise 33 to find the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And the...Ch. 14.8 - Finding a Jacobian In Exercises 35-40, And 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 Iterated 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 - Evaluating an Iterated Integral In Exercises 3-6,...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 Region In Exercises 7-10,...Ch. 14 - Finding the Area of a Region In Exercises 7-10,...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 Volume In Exercises 17-20, use a double...Ch. 14 - Finding Volume In Exercises 17-20, use a double...Ch. 14 - Average Value In Exercises 21 and 22, find the...Ch. 14 - Average Value In Exercises 21 and 22, find the...Ch. 14 - Average Temperature The temperature in degrees...Ch. 14 - Average Profit A firms profit P (in dollars) from...Ch. 14 - Converting to Polar Coordinates In Exercises 25...Ch. 14 - Converting to Polar Coordinates In Exercises 25...Ch. 14 - Volume In Exercises 27 and 28, use a double...Ch. 14 - Volume In Exercises 27 and 28, use a double...Ch. 14 - Area In Exercises 29 and 30, use a double integral...Ch. 14 - Area In Exercises 29 and 30, use a double integral...Ch. 14 - Area In Exercises 31 and 32, sketch a graph of the...Ch. 14 - Area In Exercises 31 and 32, sketch a graph of the...Ch. 14 - Area and Volume Consider the region R in the...Ch. 14 - Converting to Polar Coordinates Write the sum of...Ch. 14 - Finding the Mass of a Lamina In Exercises 35 and...Ch. 14 - Finding the Mass of a Lamina In Exercises 35 and...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 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 Gyration...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area In Exercises 43-46, find the...Ch. 14 - Finding Surface Area of the surface given by In...Ch. 14 - Building Design A new auditorium is built with a...Ch. 14 - Surface Area The roof over the stage of an open...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral In...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Volume In Exercises 55 and 56, use a triple...Ch. 14 - Volume In 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 Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral In Exercises...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Evaluating a Triple Iterated Integral Using...Ch. 14 - Volume In Exercises 67 and 68, use cylindrical...Ch. 14 - Volume In Exercises 67 and 68, use cylindrical...Ch. 14 - Volume In Exercises 69 and 70, use spherical...Ch. 14 - VolumeIn Exercises 69 and 70, use spherical...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, find the...Ch. 14 - Finding a Jacobian In Exercises 71-74, 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 - Volume Find the volume of the solid of...Ch. 14 - Surface Area Let a, b, c, and d be positive real...Ch. 14 - Using a Change of Variables The figure shows the...Ch. 14 - Proof Prove that limn0101xnyndxdy=0.Ch. 14 - Deriving a Sum Derive Eulers famous result that...Ch. 14 - Evaluating a Double Integral Evaluate the integral...Ch. 14 - Evaluating Double Integrals Evaluate the integrals...Ch. 14 - Volume Show that the volume of a spherical block...Ch. 14 - Evaluating an Integral In 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-plane...Ch. 14 - Surface Area Use the result of Exercise 14 to...Ch. 14 - Sprinkler Consider a circular lawn with a radius...Ch. 14 - Volume The figure shows a solid bounded below by...

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