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7th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337552516

Chapter 14.1, Problem 53E

Textbook Problem

Changing the Order of Integration In Exercises 51-60, sketch the region R whose area is given by the iterated integral. Then change the order of integration and show that both orders yield the same area.

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Calculus: Early Transcendental Functions

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrdCh. 14.1 - CONCEPT CHECK Region of Integration Sketch the...Ch. 14.1 - Evaluate the integral: 0x(2xy)dyCh. 14.1 - Evaluate the integral: xx2yxdyCh. 14.1 - Evaluate the integral: 04x2x2ydyCh. 14.1 - Evaluate the integral: x3x(x2+3y2)dyCh. 14.1 - Evaluate the integral: eyyylnxxdx;y0Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dxCh. 14.1 - Evaluate the integral: 0x2yeyxdyCh. 14.1 - Evaluate the integral: y2sin3xcosydx

Ch. 14.1 - Evaluate the iterated integral: 0102(x+y)dydxCh. 14.1 - Evaluate the iterated integral: 1122(x2y2)dydxCh. 14.1 - Evaluate the iterated integral: 0401ycosxdydxCh. 14.1 - Evaluate the iterated integral: 0ln40ln3ex+ydydxCh. 14.1 - Evaluate the iterated integral: 0206x2x3dydxCh. 14.1 - Evaluate the iterated integral: 010y(6x+5y3)dxdyCh. 14.1 - Evaluate the iterated integral:...Ch. 14.1 - Evaluate the iterated integral: 040x2yexdydxCh. 14.1 - Evaluate the iterated integral: 010x1x2dydxCh. 14.1 - Evaluate the iterated integral: 440x264x3dydxCh. 14.1 - Evaluate the iterated integral: 0101y2(x+y)dxdyCh. 14.1 - Evaluate the iterated integral: 023y26y2yy23ydxdyCh. 14.1 - Evaluate the iterated integral: 0204y224y2dxdyCh. 14.1 - Evaluate the iterated integral: 130y4x2+y2dxdyCh. 14.1 - Evaluate the iterated integral: 0202cosrdrdCh. 14.1 - Evaluate the iterated integral: 0433cosrdrdCh. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Evaluating an Iterated Integral In Exercises...Ch. 14.1 - Evaluating an Improper Iterated Integral In...Ch. 14.1 - Evaluating an Improper Iterated Integral In...Ch. 14.1 - Evaluate the improper iterated integral: 111xydxdyCh. 14.1 - Evaluating an Improper Iterated Integral In...Ch. 14.1 - Finding the Area of a Region In Exercises33-36,...Ch. 14.1 - Finding the Area of a Region In Exercises33-36,...Ch. 14.1 - Finding the Area of a RegionIn Exercises33-36, use...Ch. 14.1 - Finding the Area of a Region In Exercises33-36,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Finding the Area of a Region In Exercises37-42,...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - Changing the Order of Integration In Exercises...Ch. 14.1 - EXPLORING CONCEPTS Area of a Circle Write an...Ch. 14.1 - EXPLORING CONCEPTS Using Different Methods Express...Ch. 14.1 - EXPLORING CONCEPTS Think About It Determine...Ch. 14.1 - HOW DO YOU SEE IT? Use each order of integration...Ch. 14.1 - Evaluating an Iterated Integral Using Technology...Ch. 14.1 - Evaluating an Iterated Integral Using Technology...Ch. 14.1 - Evaluating an Iterated Integral Using Technology...Ch. 14.1 - Evaluating an Iterated Integral Using Technology...Ch. 14.1 - Evaluating an Iterated Integral Using TechnologyIn...Ch. 14.1 - Evaluating an Iterated Integral Using Technology...Ch. 14.1 - Comparing Different Orders of integration Using...Ch. 14.1 - Comparing Different Orders of integration Using...Ch. 14.1 - True or False? In Exercises 79 and 80, determine...Ch. 14.1 - True or False? In Exercises 79 and 80, determine...Ch. 14.2 - CONCEPT CHECK Approximating the Volume of a Solid...Ch. 14.2 - CONCEPT CHECK Fubinis Theorem What is the benefit...Ch. 14.2 - Approximation In Exercises 3-6, approximate the...Ch. 14.2 - Approximation In Exercises 3-6, approximate the...Ch. 14.2 - Approximation In Exercises 3-6, approximate the...Ch. 14.2 - Approximation In Exercises 3-6, approximate the...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises 7-12,...Ch. 14.2 - Evaluating a Double Integral In Exercises13-20,...Ch. 14.2 - Evaluating a Double Integral In Exercises13-20,...Ch. 14.2 - Evaluating a Double IntegralIn Exercises13-20, set...Ch. 14.2 - Evaluating a Double Integral In Exercises13-20,...Ch. 14.2 - Evaluating a Double IntegralIn Exercises13-20, set...Ch. 14.2 - Evaluating a Double IntegralIn Exercises13-20, set...Ch. 14.2 - Evaluating a Double IntegralIn Exercises13-20, set...Ch. 14.2 - Evaluating a Double IntegralIn Exercises13-20, set...Ch. 14.2 - Finding VolumeIn Exercise 21-26, use a double...Ch. 14.2 - Finding Volume In Exercise 21-26, use a double...Ch. 14.2 - Finding Volume In Exercise 21-26, use double...Ch. 14.2 - Finding Volume In Exercise 21-26, use double...Ch. 14.2 - Finding Volume In Exercise 21-26, use double...Ch. 14.2 - Finding Volume In Exercise 21-26, use double...Ch. 14.2 - Finding Volume In Exercises 27 and 28, use an...Ch. 14.2 - Finding Volume In Exercises 27 and 28, use an...Ch. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Finding Volume In Exercises 29-34, set up and...Ch. 14.2 - Rinding Volume In Exercises 29-34. set up and...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Volume of a Region Bounded by Two Surfaces In...Ch. 14.2 - Finding Volume Using Technology In Exercises...Ch. 14.2 - Finding Volume Using Technology In Exercises...Ch. 14.2 - Finding Volume Using Technology In Exercises...Ch. 14.2 - Finding Volume Using Technology In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Evaluating an Iterated Integral In Exercises...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Value In Exercises 51-56, find the average...Ch. 14.2 - Average Production The Cobb-Douglas production...Ch. 14.2 - Average Temperature The temperature in degrees...Ch. 14.2 - Volume Let R be a region in the xy-plane whose...Ch. 14.2 - Volume Let the plane region R be a unit circle and...Ch. 14.2 - In Exercises 61-64, show that the function is a...Ch. 14.2 - In Exercises 61-64, show that the function is a...Ch. 14.2 - In Exercises 61-64, show that the function is a...Ch. 14.2 - In Exercises 61-64, show that the function is a...Ch. 14.2 - Proof Let f be a continuous function such that...Ch. 14.2 - Finding Volume Find the volume of the solid in the...Ch. 14.2 - Approximation The table shows values of a function...Ch. 14.2 - HOW DO YOU SEE IT? The figure below shows Erie...Ch. 14.2 - True or False? In Exercises 69 and 70. determine...Ch. 14.2 - True or False? In Exercises 69 and 70. determine...Ch. 14.2 - Maximizing a Double Integral Determine the region...Ch. 14.2 - Minimizing a Double Integral Determine the region...Ch. 14.2 - Average Value Let f(x)1xet2dt. Find the average...Ch. 14.2 - Using Geometry Use a geometric argument to show...Ch. 14.2 - Evaluate 0a0bemaxb2x2,a2y2dydx where a and b are...Ch. 14.2 - Show that if 12 there does not exist a real-valued...Ch. 14.3 - CONCEPT CHECK Choosing a Coordinate System In...Ch. 14.3 - CONCEPT CHECK Choosing a Coordinate SystemIn...Ch. 14.3 - Describing Regions In your own words, describe...Ch. 14.3 - Using Polar Coordinates Sketch the region of...Ch. 14.3 - Describing a Region In Exercises 5-8, use polar...Ch. 14.3 - Describing a Region In Exercises 5-8, use polar...Ch. 14.3 - Describing a Region In Exercises 5-8, use polar...Ch. 14.3 - Describing a Region In Exercises 5-8, use polar...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral: In Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Evaluating a Double Integral in Exercises 9-16,...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar CoordinatesIn Exercises 17-26,...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises 27...Ch. 14.3 - Converting to Polar Coordinates: In Exercises 27...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - Converting to Polar Coordinates: In Exercises...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - In Exercises 33-38, use a double integral in polar...Ch. 14.3 - Volume Use a double integral in polar coordinates...Ch. 14.3 - Volume Use a double integral in polar coordinates...Ch. 14.3 - Area In Exercises 41-46, use a double integral to...Ch. 14.3 - AreaIn Exercises 41-46, use a double integral to...Ch. 14.3 - AreaIn Exercises 41-46, use a double integral to...Ch. 14.3 - AreaIn Exercises 41-46, use a double integral to...Ch. 14.3 - AreaIn Exercises 41-46, use a double integral to...Ch. 14.3 - AreaIn Exercises 41-46, use a double integral to...Ch. 14.3 - Area: In Exercises 47-52, sketch a graph of the...Ch. 14.3 - Area: In Exercises 47-52, sketch a graph of the...Ch. 14.3 - Area: In Exercises 47-52, sketch a graph of the...Ch. 14.3 - Area: In Exercises 47-52, sketch a graph of the...Ch. 14.3 - Area: In Exercises 47-52, sketch a graph of the...Ch. 14.3 - Area: In Exercises, 47-52, sketch a graph of the...Ch. 14.3 - EXPLORING CONCEPTS Area Express the area of the...Ch. 14.3 - EXPLORING CONCEPTS Comparing Integrals Let R be...Ch. 14.3 - Population The population density of a city is...Ch. 14.3 - HOW DO YOU SEE IT? Each figure shows a region of...Ch. 14.3 - Volume Determine the diameter of a hole that is...Ch. 14.3 - Glacier Horizontal cross sections of a piece of...Ch. 14.3 - Approximation In Exercises 59 and 60, use a...Ch. 14.3 - Approximation: In Exercises 59 and 60, use a...Ch. 14.3 - True or False? In Exercises 61 and 62, determine...Ch. 14.3 - True or False? In Exercises 61 and 62, determine...Ch. 14.3 - Probability The value of the integral I=ex22dx Is...Ch. 14.3 - Evaluating Integrals Use the result of Exercise 63...Ch. 14.3 - Think About It Consider the region R bounded by...Ch. 14.3 - Think About It Repeat Exercise 65 for a region R...Ch. 14.3 - Probability Find k such that the function...Ch. 14.3 - Area Show that the area of the polar sector R (see...Ch. 14.4 - CONCEPT CHECK Mass of a Planar Lamina Explain when...Ch. 14.4 - CONCEPT CHECK Moment of Inertia Describe what the...Ch. 14.4 - Finding the Mass of a Lamina In exercises 3-6,...Ch. 14.4 - Finding the Mass of a Lamina In exercises 3-6,...Ch. 14.4 - Finding the Mass of a Lamina In exercises 3-6,...Ch. 14.4 - Finding the Mass of a Lamina In exercises 3-6,...Ch. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Finding the Center of Mass In Exercises 7-10, find...Ch. 14.4 - Translations in the Plane Translate the lamina in...Ch. 14.4 - Conjecture Use the result of Exercise 11 to make a...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass In Exercises 13-24,...Ch. 14.4 - Finding the Center of Mass Using Technology In...Ch. 14.4 - Finding the Center of Mass Using Technology In...Ch. 14.4 - Finding the Center of Mass Using Technology In...Ch. 14.4 - Finding the Center of Mass Using Technology In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding the Radius of Gyration About Each Axis In...Ch. 14.4 - Finding Moments of Inertia and Radii of Gyration...Ch. 14.4 - Finding Moments of Inertia and Radii of Gyration...Ch. 14.4 - Finding Moments of Inertia and Radii of Gyration...Ch. 14.4 - Finding Moments of Inertia and Radii of Gyration...Ch. 14.4 - Finding a Moment of Inertia Using Technology In...Ch. 14.4 - Finding a Moment of Inertia Using Technology In...Ch. 14.4 - Finding a Moment of Inertia Using Technology In...Ch. 14.4 - Finding a Moment of Inertia Using Technology In...Ch. 14.4 - Hydraulics In Exercises 43-46, determine the...Ch. 14.4 - Hydraulics In Exercises 43-46, determine the...Ch. 14.4 - Hydraulics In Exercises 43-46, determine the...Ch. 14.4 - Hydraulics In Exercises 43-46, determine the...Ch. 14.4 - Polar Moment of Inertia What does it mean for an...Ch. 14.4 - HOW DO YOU SEE IT? The center of mass of the...Ch. 14.4 - Proof Prove the following Theorem of Pappus: Let R...Ch. 14.5 - CONCEPT CHECK Surface Area What is the...Ch. 14.5 - CONCEPT CHECK Numerical Integration Write a double...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Areathe area of the surface given...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding SurfaceArea In Exercises 3-16, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 3-16, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area In Exercises 17-20, find the...Ch. 14.5 - Finding Surface AreaIn Exercises 17-20, find the...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using TechnologyIn...Ch. 14.5 - Finding Surface Area Using Technology In Exercises...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - Setting Up a Double IntegralIn Exercises 27-30,...Ch. 14.5 - EXPLORING CONCEPTS Surface Area Will the surface...Ch. 14.5 - HOW DO YOU SEE IT? 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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