Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)
Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)
3rd Edition
ISBN: 9780134770512
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Concept explainers

Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
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Chapter 16.5, Problem 72E

Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables.

70.    Volume of a drilled hemisphere Find the volume of material remaining in a hemisphere of radius 2 after a cylindrical hole of radius 1 is drilled through the center of the hemisphere perpendicular to its base.

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Chapter 16 Solutions

Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)

Ch. 16.1 - Explain why the sum for the volume is an...Ch. 16.1 - Consider the integral 3412f(x,y)dxdy. Give the...Ch. 16.1 - Prob. 3QCCh. 16.1 - 1. Write an iterated integral that gives the...Ch. 16.1 - Write an iterated integral that gives the volume...Ch. 16.1 - Write two iterated integrals that equal Rf(x,y)dA,...Ch. 16.1 - Consider the integral 1311(2y2+xy)dydx. State the...Ch. 16.1 - Region R = {(x, y) : 0 ≤ x ≤ 4, 0 ≤ y ≤ 6} is...Ch. 16.1 - Pictures of solids Draw the solid whose volume is...Ch. 16.1 - Iterated integrals Evaluate the following iterated...
Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - More integration practice Evaluate the following...Ch. 16.1 - More integration practice Evaluate the following...Ch. 16.1 - More integration practice Evaluate the following...Ch. 16.1 - Iterated integrals Evaluate the following iterated...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Double integrals Evaluate each double integral...Ch. 16.1 - Volumes of solids Find the volume of the following...Ch. 16.1 - Volumes of solids Find the volume of the following...Ch. 16.1 - Volumes of solids Find the volume of the following...Ch. 16.1 - Volumes of solids Find the volume of the following...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Choose a convenient order When convened to an...Ch. 16.1 - Average value Compute the average value of the...Ch. 16.1 - Average value Compute the average value of the...Ch. 16.1 - Average value Compute the average value of the...Ch. 16.1 - Average value 35.Find the average squared distance...Ch. 16.1 - Average value 35.Find the average squared distance...Ch. 16.1 - Explain why or why not Determine whether the...Ch. 16.1 - Symmetry Evaluate the following integrals using...Ch. 16.1 - Computing populations The population densities in...Ch. 16.1 - Prob. 54ECh. 16.1 - Cylinders Let S be the solid in 3between the...Ch. 16.1 - Product of integrals Suppose f(x, y) = g(x)h(y),...Ch. 16.1 - Solving for a parameter Let R={x,y}:{0x,0ya}. For...Ch. 16.1 - Prob. 58ECh. 16.1 - Zero average value Find the value of a 0 such...Ch. 16.1 - Prob. 60ECh. 16.1 - Density and mass Suppose a thin rectangular plate,...Ch. 16.1 - Approximating volume Propose a method based on...Ch. 16.1 - Prob. 63ECh. 16.2 - A region R is bounded by the x- and y-axes and the...Ch. 16.2 - Could the integral in Example 2 be evaluated by...Ch. 16.2 - Change the order of integration of the integral...Ch. 16.2 - Prob. 4QCCh. 16.2 - Describe and sketch a region that is bounded above...Ch. 16.2 - Describe and a sketch a region that is bounded on...Ch. 16.2 - Which order of integration is preferable to...Ch. 16.2 - Which order of integration would you use to find...Ch. 16.2 - Change the order of integration in the integral...Ch. 16.2 - Sketch the region of integration for 22x24exydydxCh. 16.2 - Sketch the region of integration for 0202x dy dx...Ch. 16.2 - Describe a solid whose volume equals 4416z216z2 10...Ch. 16.2 - Regions of integration Consider the regions R...Ch. 16.2 - Regions of integration Consider the regions R...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Sketch each region and...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Regions of integration Write an iterated integral...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Prob. 52ECh. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Evaluating integrals Evaluate the following...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Changing order of integration Reverse the order of...Ch. 16.2 - Two integrals to one Draw the regions of...Ch. 16.2 - Two integrals to one Draw the regions of...Ch. 16.2 - Volumes Find the volume of the following solids....Ch. 16.2 - Volumes Find the volume of the following solids....Ch. 16.2 - Volumes Use double integrals to calculate the...Ch. 16.2 - Volumes Use double integrals to calculate the...Ch. 16.2 - Volumes Use double integrals to calculate the...Ch. 16.2 - Regions between surfaces Find the volume of the...Ch. 16.2 - Regions between surfaces Find the volume of the...Ch. 16.2 - Volumes Find the volume of the following solids....Ch. 16.2 - Regions between surfaces Find the volume of the...Ch. 16.2 - Regions between surfaces Find the volume of the...Ch. 16.2 - Volume using technology Find the volume of the...Ch. 16.2 - Volume using technology Find the volume of the...Ch. 16.2 - Volume using technology Find the volume of the...Ch. 16.2 - Volume using technology Find the volume of the...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Area of plane regions Use double integrals to...Ch. 16.2 - Explain why or why not Determine whether the...Ch. 16.2 - Related integrals Evaluate each integral a....Ch. 16.2 - Volumes Compute the volume of the following...Ch. 16.2 - Square region Consider the region R = {(x, y): |x|...Ch. 16.2 - Average value Use the definition for the average...Ch. 16.2 - Average value Use the definition for the average...Ch. 16.2 - Area integrals Consider the following regions R....Ch. 16.2 - Area integrals Consider the following regions R....Ch. 16.2 - Improper integrals Many improper double integrals...Ch. 16.2 - Prob. 100ECh. 16.2 - Improper integrals Many improper double integrals...Ch. 16.2 - Prob. 102ECh. 16.3 - Describe in polar coordinates the region in the...Ch. 16.3 - Express the functions f(x, y) = (x2 + y2)5/2 and...Ch. 16.3 - Give a geometric explanation for the extraneous...Ch. 16.3 - Express the area of the disk R ={(r, ) : 0 r a,...Ch. 16.3 - Draw the region {(r, ): 1 r 2, 0 /2}. Why is...Ch. 16.3 - Write the double integral Rf(x,y)dAas an iterated...Ch. 16.3 - Sketch in the xy-plane the region of integration...Ch. 16.3 - Prob. 4ECh. 16.3 - How do you find the area of a region R = {(r, ):...Ch. 16.3 - How do you find the average value of a function...Ch. 16.3 - Polar rectangles Sketch the following polar...Ch. 16.3 - Polar rectangles Sketch the following polar...Ch. 16.3 - Polar rectangles Sketch the following polar...Ch. 16.3 - Polar rectangles Sketch the following polar...Ch. 16.3 - Volume of solids Find the volume of the solid...Ch. 16.3 - Volume of solids Find the volume of the solid...Ch. 16.3 - Volume of solids Find the volume of the solid...Ch. 16.3 - Volume of solids Find the volume of the solid...Ch. 16.3 - Solids bounded by paraboloids Find the volume of...Ch. 16.3 - Solids bounded by paraboloids Find the volume of...Ch. 16.3 - Solids bounded by paraboloids Find the volume of...Ch. 16.3 - Solids bounded by paraboloids Find the volume of...Ch. 16.3 - Solids bounded by hyperboloids Find the volume of...Ch. 16.3 - Solids bounded by hyperboloids Find the volume of...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Sketch the given...Ch. 16.3 - Cartesian to polar coordinates Evaluate the...Ch. 16.3 - Cartesian to polar coordinates Evaluate the...Ch. 16.3 - Cartesian to polar coordinates Evaluate the...Ch. 16.3 - Cartesian to polar coordinates Evaluate the...Ch. 16.3 - Regions between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Volume between surfaces Find the volume of the...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Describing general regions Sketch the following...Ch. 16.3 - Computing areas Use a double integral to find the...Ch. 16.3 - Computing areas Use a double integral to find the...Ch. 16.3 - Computing areas Use a double integral to find the...Ch. 16.3 - Computing areas Use a double integral to find the...Ch. 16.3 - 47-52. Computing areas Use a double integral to...Ch. 16.3 - Computing areas Use a double integral to find the...Ch. 16.3 - Average values Find the following average values....Ch. 16.3 - Average values Find the following average values....Ch. 16.3 - Explain why or why not Determine whether the...Ch. 16.3 - Areas of circles Use integration to show that the...Ch. 16.3 - Filling bowls with water Which bowl holds more...Ch. 16.3 - Equal volumes To what height (above the bottom of...Ch. 16.3 - Volume of a hyperbolic paraboloid Consider the...Ch. 16.3 - Volume of a sphere Use double integrals in polar...Ch. 16.3 - Volume Find the volume of the solid bounded by the...Ch. 16.3 - Volume Find the volume of the solid bounded by the...Ch. 16.3 - Miscellaneous integrals Evaluate the following...Ch. 16.3 - Miscellaneous integrals Evaluate the following...Ch. 16.3 - Improper integrals Improper integrals arise in...Ch. 16.3 - Improper integrals Improper integrals arise in...Ch. 16.3 - Improper integrals Improper integrals arise in...Ch. 16.3 - Improper integrals Improper integrals arise in...Ch. 16.3 - Prob. 69ECh. 16.3 - Mass from density data The following table gives...Ch. 16.3 - A mass calculation Suppose the density of a thin...Ch. 16.3 - Area formula In Section 12.3 it was shown that the...Ch. 16.3 - Normal distribution An important integral in...Ch. 16.3 - Existence of integrals For what values of p does...Ch. 16.3 - Integrals in strips Consider the integral...Ch. 16.4 - List the six orders in which the three...Ch. 16.4 - Write the integral in Example 1 in the orders dx...Ch. 16.4 - Write the integral in Example 2 in the orders dz...Ch. 16.4 - Without integrating, what is the average value of...Ch. 16.4 - Sketch the region D = {(x, y, z): x2 + y2 4, 0 z...Ch. 16.4 - Write an iterated integral for Df(x,y,z)dV, where...Ch. 16.4 - Write an iterated integral for Df(x,y,z)dV, where...Ch. 16.4 - Sketch the region of integration for the integral...Ch. 16.4 - Write the integral in Exercise 4 in the order dy...Ch. 16.4 - Write an integral for the average value of f(x, y,...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Integrals over boxes Evaluate the following...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Volumes of solids Use a triple integral to find...Ch. 16.4 - Volumes of solids Use a triple integral to find...Ch. 16.4 - Volumes of solids Use a triple integral to find...Ch. 16.4 - Finding an appropriate order of integration Find...Ch. 16.4 - Finding an appropriate order of integration Find...Ch. 16.4 - Finding an appropriate order of integration Find...Ch. 16.4 - Finding an appropriate order of integration Find...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - Six orderings Let D be the solid in the first...Ch. 16.4 - All six orders Let D be the solid bounded by y =...Ch. 16.4 - Changing order of integration Write the integral...Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Triple integrals Evaluate the following integrals....Ch. 16.4 - Changing the order of integration Rewrite the...Ch. 16.4 - Changing the order of integration Rewrite the...Ch. 16.4 - Changing the order of integration Rewrite the...Ch. 16.4 - Changing the order of integration Rewrite the...Ch. 16.4 - Average value Find the following average values....Ch. 16.4 - Average value Find the following average values....Ch. 16.4 - Average value Find the following average values....Ch. 16.4 - Average value Find the following average values....Ch. 16.4 - Explain why or why not Determine whether the...Ch. 16.4 - Changing the order of integration Use another...Ch. 16.4 - Prob. 57ECh. 16.4 - Miscellaneous volumes Use a triple integral to...Ch. 16.4 - Prob. 59ECh. 16.4 - Prob. 60ECh. 16.4 - Miscellaneous volumes Use a triple integral to...Ch. 16.4 - Volumes of solids. Find the volume of the...Ch. 16.4 - Two cylinders The x- and y-axes form the axes of...Ch. 16.4 - Three cylinders The coordinate axes form the axes...Ch. 16.4 - Dividing the cheese Suppose a wedge of cheese...Ch. 16.4 - Partitioning a cube Consider the region D1 = {(x,...Ch. 16.4 - General volume formulas Find equations for the...Ch. 16.4 - Prob. 68ECh. 16.4 - General volume formulas Find equations for the...Ch. 16.4 - Prob. 70ECh. 16.4 - Prob. 71ECh. 16.4 - Prob. 72ECh. 16.4 - Hypervolume Find the Volume of the...Ch. 16.4 - Prob. 74ECh. 16.5 - Prob. 1QCCh. 16.5 - Find the limits of integration for a triple...Ch. 16.5 - Find the spherical coordinates of the point with...Ch. 16.5 - Explain how cylindrical coordinates are used to...Ch. 16.5 - Explain how spherical coordinates are used to...Ch. 16.5 - Describe the set {(r, , z): r = 4z} in cylindrical...Ch. 16.5 - Describe the set {(, , ): = /4} in spherical...Ch. 16.5 - Explain why dz r dr d is the volume of a small box...Ch. 16.5 - Explain why 2 sin d d d is the volume of a small...Ch. 16.5 - Write the integral D w(r, , z) dV as an iterated...Ch. 16.5 - Write the integral D w(, , ) dV as an interated...Ch. 16.5 - What coordinate system is suggested if the...Ch. 16.5 - What coordinate system is suggested if the...Ch. 16.5 - Sets in cylindrical coordinates Identify and...Ch. 16.5 - Sets in cylindrical coordinates Identify and...Ch. 16.5 - Sets in cylindrical coordinates Identify and...Ch. 16.5 - Sets in cylindrical coordinates Identify and...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Which weighs more? For 0 r 1, the solid bounded...Ch. 16.5 - Prob. 28ECh. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 16.5 - Prob. 35ECh. 16.5 - Sets in spherical coordinates Identify and sketch...Ch. 16.5 - Sets in spherical coordinates Identify and sketch...Ch. 16.5 - Sets in spherical coordinates Identify and sketch...Ch. 16.5 - Seattle has a latitude of 47.6° North and a...Ch. 16.5 - Los Angeles has a latitude of 34.05* North and a...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Integrals in spherical coordinates Evaluate the...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Volumes in spherical coordinates Use spherical...Ch. 16.5 - Explain why or why not Determine whether the...Ch. 16.5 - Spherical to rectangular Convert the equation 2 =...Ch. 16.5 - Spherical to rectangular Convert the equation 2 =...Ch. 16.5 - Prob. 58ECh. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Mass from density Find the mass of the following...Ch. 16.5 - Changing order of integration If possible, write...Ch. 16.5 - Changing order of integration If possible, write...Ch. 16.5 - Changing order of integration if possible, write...Ch. 16.5 - Changing order of integration if possible, write...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Miscellaneous volumes Choose the best coordinate...Ch. 16.5 - Density distribution A right circular cylinder...Ch. 16.5 - Charge distribution A spherical cloud of electric...Ch. 16.5 - Gravitational field due to spherical shell A point...Ch. 16.5 - Water in a gas tank Before a gasoline-powered...Ch. 16.5 - General volume formulas Use integration to find...Ch. 16.5 - Prob. 78ECh. 16.5 - General volume formulas Use integration to find...Ch. 16.5 - Prob. 80ECh. 16.5 - Intersecting spheres One sphere is centered at the...Ch. 16.6 - A 90-kg person sits 2 m from the balance point of...Ch. 16.6 - Solve the equation m1(x1x)+m2(x2x)=0 for x to...Ch. 16.6 - Prob. 3QCCh. 16.6 - Explain why the integral for My has x in the...Ch. 16.6 - Prob. 5QCCh. 16.6 - Explain how to find the balance point for two...Ch. 16.6 - If a thin 1-m cylindrical rod has a density of =...Ch. 16.6 - Explain how to find the center of mass of a thin...Ch. 16.6 - In the integral for the moment Mx of a thin plate,...Ch. 16.6 - Explain how to find the center of mass of a...Ch. 16.6 - In the integral for the moment Mxz with respect to...Ch. 16.6 - Individual masses on a line Sketch the following...Ch. 16.6 - Individual masses on a line Sketch the following...Ch. 16.6 - One-dimensional objects Find the mass and center...Ch. 16.6 - One-dimensional objects Find the mass and center...Ch. 16.6 - One-dimensional objects Find the mass and center...Ch. 16.6 - One-dimensional objects Find the mass and center...Ch. 16.6 - One-dimensional objects Find the mass and center...Ch. 16.6 - Prob. 14ECh. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Centroid calculations Find the mass and centroid...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Variable-density plates Find the center of mass of...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Center of mass of constant-density solids Find the...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Variable-density solids Find the coordinates of...Ch. 16.6 - Explain why or why not Determine whether the...Ch. 16.6 - Limiting center of mass A thin rod of length L has...Ch. 16.6 - Limiting center of mass A thin rod of length L has...Ch. 16.6 - Prob. 42ECh. 16.6 - Two-dimensional plates Find the mass and center of...Ch. 16.6 - Two-dimensional plates Find the mass and center of...Ch. 16.6 - Centroids Use polar coordinates to find the...Ch. 16.6 - Prob. 46ECh. 16.6 - Centroids Use polar coordinates to find the...Ch. 16.6 - Centroids Use polar coordinates to find the...Ch. 16.6 - Prob. 49ECh. 16.6 - Prob. 50ECh. 16.6 - Prob. 51ECh. 16.6 - Prob. 52ECh. 16.6 - Prob. 53ECh. 16.6 - Prob. 54ECh. 16.6 - Centers of mass for general objects Consider the...Ch. 16.6 - Centers of mass for general objects Consider the...Ch. 16.6 - Prob. 57ECh. 16.6 - Centers of mass for general objects Consider the...Ch. 16.6 - Centers of mass for general objects Consider the...Ch. 16.6 - Geographic vs. population center Geographers...Ch. 16.6 - Center of mass on the edge Consider the thin...Ch. 16.6 - Center of mass on the edge Consider the...Ch. 16.6 - Draining a soda can A cylindrical soda can has a...Ch. 16.6 - Triangle medians A triangular region has a base...Ch. 16.6 - The golden earring A disk of radius r is removed...Ch. 16.7 - Prob. 1QCCh. 16.7 - Prob. 2QCCh. 16.7 - Solve the equations u = x + y, v = –x + 2y for x...Ch. 16.7 - Prob. 4QCCh. 16.7 - Interpret a Jacobian with a value of 1 (as in...Ch. 16.7 - Suppose S is the unit square in the first quadrant...Ch. 16.7 - Explain how to compute the Jacobian of the...Ch. 16.7 - Using the transformation T: x = u + v, y = u v,...Ch. 16.7 - Suppose S is the unit cube in the first octant of...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 16.7 - Images of regions Find the image R in the xy-plane...Ch. 16.7 - Images of regions Find the image R in the xy-plane...Ch. 16.7 - Images of regions Find the image R in the xy-plane...Ch. 16.7 - Images of regions Find the image R in the xy-plane...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16.7 - Solve and compute Jacobians Solve the following...Ch. 16.7 - Solve and compute Jacobians Solve the following...Ch. 16.7 - Solve and compute Jacobians Solve the following...Ch. 16.7 - Solve and compute Jacobians Solve the following...Ch. 16.7 - Double integralstransformation given To evaluate...Ch. 16.7 - Double integralstransformation given To evaluate...Ch. 16.7 - Double integralstransformation given To evaluate...Ch. 16.7 - Double integralstransformation given To evaluate...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Double integralsyour choice of transformation...Ch. 16.7 - Jacobians in three variables Evaluate the...Ch. 16.7 - Prob. 38ECh. 16.7 - Jacobians in three variables Evaluate the...Ch. 16.7 - Jacobians in three variables Evaluate the...Ch. 16.7 - Triple integrals Use a change of variables to...Ch. 16.7 - Triple integrals Use a change of variables to...Ch. 16.7 - Triple integrals Use a change of variables to...Ch. 16.7 - Triple integrals Use a change of variables to...Ch. 16.7 - Explain why or why not Determine whether the...Ch. 16.7 - Prob. 46ECh. 16.7 - Prob. 47ECh. 16.7 - Ellipse problems Let R be the region bounded by...Ch. 16.7 - Ellipse problems Let R be the region bounded by...Ch. 16.7 - Ellipse problems Let R be the region bounded by...Ch. 16.7 - Ellipse problems Let R be the region bounded by...Ch. 16.7 - Ellipse problems Let R be the region bounded by...Ch. 16.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 16.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 16.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 16.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 16.7 - Parabolic coordinates Let T be the transformation...Ch. 16.7 - Shear transformations in 3 The transformation T in...Ch. 16.7 - Linear transformations Consider the linear...Ch. 16.7 - Meaning of the Jacobian The Jacobian is a...Ch. 16.7 - Open and closed boxes Consider the region R...Ch. 16 - Explain why or why not Determine whether the...Ch. 16 - Evaluating integrals Evaluate the following...Ch. 16 - Evaluating integrals Evaluate the following...Ch. 16 - Evaluating integrals Evaluate the following...Ch. 16 - Changing the order of integration Assuming f is...Ch. 16 - Changing the order of integration Assuming f is...Ch. 16 - Changing the order of integration Assuming f is...Ch. 16 - Area of plane regions Use double integrals to...Ch. 16 - Area of plane regions Use double integrals to...Ch. 16 - Area of plane regions Use double integrals to...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Miscellaneous double integrals Choose a convenient...Ch. 16 - Cartesian to polar coordinates Evaluate the...Ch. 16 - Cartesian to polar coordinates Evaluate the...Ch. 16 - Computing areas Sketch the following regions and...Ch. 16 - Computing areas Sketch the following regions and...Ch. 16 - Computing areas Sketch the following regions and...Ch. 16 - Average values 22.Find the average value of...Ch. 16 - Average values 23.Find the average distance from...Ch. 16 - Changing order of integration Rewrite the...Ch. 16 - Changing order of integration Rewrite the...Ch. 16 - Changing order of integration Rewrite the...Ch. 16 - Triple integrals Evaluate the following integrals,...Ch. 16 - Triple integrals Evaluate the following integrals,...Ch. 16 - Triple integrals Evaluate the following integrals,...Ch. 16 - Triple integrals Evaluate the following integrals,...Ch. 16 - Triple integrals Evaluate the following integrals,...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Volumes of solids Find the volume of the following...Ch. 16 - Single to double integral Evaluate...Ch. 16 - Tetrahedron limits Let D be the tetrahedron with...Ch. 16 - Polar to Cartesian Evaluate using rectangular...Ch. 16 - Average value 40.Find the average of the square of...Ch. 16 - Average value 41.Find the average x-coordinate of...Ch. 16 - Integrals in cylindrical coordinates Evaluate the...Ch. 16 - Integrals in cylindrical coordinates Evaluate the...Ch. 16 - Volumes in cylindrical coordinates Use integration...Ch. 16 - Volumes in cylindrical coordinates Use integration...Ch. 16 - Integrals in spherical coordinates Evaluate the...Ch. 16 - Integrals in spherical coordinates Evaluate the...Ch. 16 - Volumes in spherical coordinates Use integration...Ch. 16 - Volumes in spherical coordinates Use integration...Ch. 16 - Volumes in spherical coordinates Use integration...Ch. 16 - Center of mass of constant-density plates Find the...Ch. 16 - Center of mass of constant-density plates Find the...Ch. 16 - Center of mass of constant-density plates Find the...Ch. 16 - Center of mass of constant-density plates Find the...Ch. 16 - Center of mass of constant-density solids Find the...Ch. 16 - Center of mass of constant-density solids Find the...Ch. 16 - Prob. 59RECh. 16 - Variable-density solids Find the coordinates of...Ch. 16 - Center of mass for general objects Consider the...Ch. 16 - Prob. 62RECh. 16 - A sector of a circle in the first quadrant is...Ch. 16 - An ice cream cone is bounded above by the sphere...Ch. 16 - Volume and weight of a fish tank A spherical fish...Ch. 16 - Prob. 67RECh. 16 - Transforming a square Let S = {(u, v): 0 u 1, 0...Ch. 16 - 67–70. Transforming a square Let S = {(u, v) : 0 ≤...Ch. 16 - Transforming a square Let S = {(u, v): 0 u 1, 0...Ch. 16 - Prob. 71RECh. 16 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 16 - Prob. 73RECh. 16 - Prob. 74RECh. 16 - Double integralstransformation given To evaluate...Ch. 16 - Double integralstransformation given To evaluate...Ch. 16 - 75-78. Double integrals—transformation given To...Ch. 16 - Double integralstransformation given To evaluate...Ch. 16 - Double integrals Evaluate the following integrals...Ch. 16 - Double integrals Evaluate the following integrals...Ch. 16 - Triple integrals Use a change of variables to...Ch. 16 - Triple integrals Use a change of variables to...

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