Calculus: Early Transcendentals, 2nd Edition
2nd Edition
ISBN: 9780321965165
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Textbook Question
Chapter 13.3, Problem 43E
Computing areas Sketch each region and use a double
43. The region inside both the cardioids r = 1 + sin θ and the cardioid r = 1 + cos θ
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Chapter 13 Solutions
Calculus: Early Transcendentals, 2nd Edition
Ch. 13.1 - 1. Write an iterated integral that gives the...Ch. 13.1 - Write an iterated integral that gives the volume...Ch. 13.1 - Write two iterated integrals that equal Rf(x,y)dA,...Ch. 13.1 - Consider the integral 1311(2y2+xy)dydx. State the...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...
Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Prob. 12ECh. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Iterated integrals Evaluate the following iterated...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Prob. 20ECh. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Double integrals Evaluate each double integral...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Choose a convenient order When convened to an...Ch. 13.1 - Average value Compute the average value of the...Ch. 13.1 - Average value Compute the average value of the...Ch. 13.1 - Average value Compute the average value of the...Ch. 13.1 - Average value 35.Find the average squared distance...Ch. 13.1 - Average value 35.Find the average squared distance...Ch. 13.1 - Explain why or why not Determine whether the...Ch. 13.1 - Symmetry Evaluate the following integrals using...Ch. 13.1 - Computing populations The population densities in...Ch. 13.1 - Prob. 40ECh. 13.1 - Pictures of solids Draw the solid whose volume is...Ch. 13.1 - Prob. 42ECh. 13.1 - More integration practice Evaluate the following...Ch. 13.1 - More integration practice Evaluate the following...Ch. 13.1 - More integration practice Evaluate the following...Ch. 13.1 - More integration practice Evaluate the following...Ch. 13.1 - Volumes of solids Find the volume of the following...Ch. 13.1 - Prob. 48ECh. 13.1 - Volumes of solids Find the volume of the following...Ch. 13.1 - Prob. 50ECh. 13.1 - Solving for a parameter Let R={x,y}:{0x,0ya}. For...Ch. 13.1 - Prob. 52ECh. 13.1 - Zero average value Find the value of a 0 such...Ch. 13.1 - Prob. 54ECh. 13.1 - Density and mass Suppose a thin rectangular plate,...Ch. 13.1 - Approximating volume Propose a method based on...Ch. 13.1 - Cylinders Let S be the solid in 3between the...Ch. 13.1 - Product of integrals Suppose f(x, y) = g(x)h(y),...Ch. 13.1 - Prob. 59ECh. 13.1 - Prob. 60ECh. 13.1 - A generalization Let R be as in Exercise 60, let F...Ch. 13.2 - Describe and sketch a region that is bounded above...Ch. 13.2 - Describe and a sketch a region that is bounded on...Ch. 13.2 - Which order of integration is preferable to...Ch. 13.2 - Which order of integration would you use to find...Ch. 13.2 - Change the order of integration in the integral...Ch. 13.2 - Sketch the region of integration for 22x24exydydxCh. 13.2 - Regions of integration Consider the regions R...Ch. 13.2 - Regions of integration Consider the regions R...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Prob. 20ECh. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Prob. 22ECh. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Regions of integration Write an iterated integral...Ch. 13.2 - Regions of integration Write an iterated integral...Ch. 13.2 - Prob. 33ECh. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Regions of integration Sketch each region and...Ch. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - Prob. 39ECh. 13.2 - Prob. 40ECh. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Prob. 42ECh. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Evaluating integrals Evaluate the following...Ch. 13.2 - Prob. 52ECh. 13.2 - Volumes Use double integrals to calculate the...Ch. 13.2 - Volumes Use double integrals to calculate the...Ch. 13.2 - Volumes Use double integrals to calculate the...Ch. 13.2 - Prob. 56ECh. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Changing order of integration Reverse the order of...Ch. 13.2 - Prob. 63ECh. 13.2 - Changing order of integration The following...Ch. 13.2 - Prob. 65ECh. 13.2 - Changing order of integration The following...Ch. 13.2 - Prob. 67ECh. 13.2 - Changing order of integration The following...Ch. 13.2 - Regions between surfaces Find the volume of the...Ch. 13.2 - Prob. 70ECh. 13.2 - Regions between surfaces Find the volume of the...Ch. 13.2 - Regions between surfaces Find the volume of the...Ch. 13.2 - Regions between surfaces Find the volume of the...Ch. 13.2 - Regions between surfaces Find the volume of the...Ch. 13.2 - Prob. 75ECh. 13.2 - Prob. 76ECh. 13.2 - Prob. 77ECh. 13.2 - Prob. 78ECh. 13.2 - Prob. 79ECh. 13.2 - Prob. 80ECh. 13.2 - Explain why or why not Determine whether the...Ch. 13.2 - Prob. 82ECh. 13.2 - Prob. 83ECh. 13.2 - Prob. 84ECh. 13.2 - Prob. 85ECh. 13.2 - Prob. 86ECh. 13.2 - Two integrals to one Draw the regions of...Ch. 13.2 - Square region Consider the region R = {(x, y): |x|...Ch. 13.2 - Prob. 89ECh. 13.2 - Prob. 90ECh. 13.2 - Area integrals Consider the following regions R....Ch. 13.2 - Area integrals Consider the following regions R....Ch. 13.2 - Prob. 93ECh. 13.2 - Prob. 94ECh. 13.2 - Prob. 95ECh. 13.2 - Prob. 96ECh. 13.2 - Volumes Compute the volume of the following...Ch. 13.2 - Prob. 99ECh. 13.2 - Prob. 100ECh. 13.2 - Prob. 101ECh. 13.2 - Prob. 102ECh. 13.2 - Prob. 103ECh. 13.3 - Draw the region {(r, ): 1 r 2, 0 /2}. Why is...Ch. 13.3 - Write the double integral Rf(x,y)dAas an iterated...Ch. 13.3 - Sketch the region of integration for the integral...Ch. 13.3 - Prob. 4ECh. 13.3 - How do you find the area of a region R = {(r, ):...Ch. 13.3 - How do you find the average value of a function...Ch. 13.3 - Polar rectangles Sketch the following polar...Ch. 13.3 - Polar rectangles Sketch the following polar...Ch. 13.3 - Polar rectangles Sketch the following polar...Ch. 13.3 - Polar rectangles Sketch the following polar...Ch. 13.3 - Solids bounded by paraboloids Find the volume of...Ch. 13.3 - Solids bounded by paraboloids Find the volume of...Ch. 13.3 - Solids bounded by paraboloids Find the volume of...Ch. 13.3 - Solids bounded by paraboloids Find the volume of...Ch. 13.3 - Solids bounded by hyperboloids Find the volume of...Ch. 13.3 - Solids bounded by hyperboloids Find the volume of...Ch. 13.3 - Solids bounded by hyperboloids Find the volume of...Ch. 13.3 - Solids bounded by hyperboloids Find the volume of...Ch. 13.3 - Volume between surfaces Find the volume of the...Ch. 13.3 - Volume between surfaces Find the volume of the...Ch. 13.3 - Volume between surfaces Find the volume of the...Ch. 13.3 - Prob. 22ECh. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Cartesian to polar coordinates Sketch the given...Ch. 13.3 - Island problems The surface of an island is...Ch. 13.3 - Island problems The surface of an island is...Ch. 13.3 - Island problems The surface of an island is...Ch. 13.3 - Prob. 32ECh. 13.3 - Describing general regions Sketch the following...Ch. 13.3 - Prob. 34ECh. 13.3 - Describing general regions Sketch the following...Ch. 13.3 - Describing general regions Sketch the following...Ch. 13.3 - Describing general regions Sketch the following...Ch. 13.3 - Describing general regions Sketch the following...Ch. 13.3 - Computing areas Sketch each region and use a...Ch. 13.3 - Computing areas Sketch each region and use a...Ch. 13.3 - Computing areas Sketch each region and use a...Ch. 13.3 - Computing areas Sketch each region and use a...Ch. 13.3 - Computing areas Sketch each region and use a...Ch. 13.3 - Prob. 44ECh. 13.3 - Average values Find the following average values....Ch. 13.3 - Prob. 46ECh. 13.3 - Prob. 47ECh. 13.3 - Average values Find the following average values....Ch. 13.3 - Explain why or why not Determine whether the...Ch. 13.3 - Prob. 50ECh. 13.3 - Miscellaneous integrals Evaluate the following...Ch. 13.3 - Miscellaneous integrals Evaluate the following...Ch. 13.3 - Miscellaneous integrals Evaluate the following...Ch. 13.3 - Prob. 54ECh. 13.3 - Prob. 55ECh. 13.3 - Miscellaneous integrals Evaluate the following...Ch. 13.3 - Miscellaneous integrals Evaluate the following...Ch. 13.3 - Areas of circles Use integration to show that the...Ch. 13.3 - Filling bowls with water Which bowl holds more...Ch. 13.3 - Prob. 60ECh. 13.3 - Volume of a hyperbolic paraboloid Consider the...Ch. 13.3 - Prob. 62ECh. 13.3 - Improper integrals Improper integrals arise in...Ch. 13.3 - Improper integrals Improper integrals arise in...Ch. 13.3 - Improper integrals Improper integrals arise in...Ch. 13.3 - Improper integrals Improper integrals arise in...Ch. 13.3 - Prob. 67ECh. 13.3 - Mass from density data The following table gives...Ch. 13.3 - A mass calculation Suppose the density of a thin...Ch. 13.3 - Prob. 70ECh. 13.3 - Prob. 71ECh. 13.3 - Existence of integrals For what values of p does...Ch. 13.3 - Integrals in strips Consider the integral...Ch. 13.3 - Prob. 74ECh. 13.4 - Sketch the region D = {(x, y, z): x2 + y2 4, 0 z...Ch. 13.4 - Write an iterated integral for Df(x,y,z)dV, where...Ch. 13.4 - Write an iterated integral for Df(x,y,z)dV, where...Ch. 13.4 - Sketch the region of integration for the integral...Ch. 13.4 - Write the integral in Exercise 4 in the order dy...Ch. 13.4 - Write an integral for the average value of f(x, y,...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Integrals over boxes Evaluate the following...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Volumes of solids. Find the volume of the...Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Triple integrals Evaluate the following integrals....Ch. 13.4 - Prob. 34ECh. 13.4 - Finding an appropriate order of integration Find...Ch. 13.4 - Finding an appropriate order of integration Find...Ch. 13.4 - Finding an appropriate order of integration Find...Ch. 13.4 - Finding an appropriate order of integration Find...Ch. 13.4 - Changing the order of integration Rewrite the...Ch. 13.4 - Changing the order of integration Rewrite the...Ch. 13.4 - Changing the order of integration Rewrite the...Ch. 13.4 - Changing the order of integration Rewrite the...Ch. 13.4 - Average value Find the following average values....Ch. 13.4 - Prob. 44ECh. 13.4 - Average value Find the following average values....Ch. 13.4 - Prob. 46ECh. 13.4 - Average value Find the following average values....Ch. 13.4 - Prob. 48ECh. 13.4 - Explain why or why not Determine whether the...Ch. 13.4 - Changing the order of integration Use another...Ch. 13.4 - Prob. 51ECh. 13.4 - Prob. 52ECh. 13.4 - Miscellaneous volumes Use a triple integral to...Ch. 13.4 - Miscellaneous volumes Use a triple integral to...Ch. 13.4 - Prob. 55ECh. 13.4 - Prob. 56ECh. 13.4 - Changing order of integration Write the integral...Ch. 13.4 - All six orders Let D be the solid bounded by y =...Ch. 13.4 - Prob. 59ECh. 13.4 - Dividing the cheese Suppose a wedge of cheese...Ch. 13.4 - General volume formulas Find equations for the...Ch. 13.4 - Prob. 62ECh. 13.4 - General volume formulas Find equations for the...Ch. 13.4 - Prob. 64ECh. 13.4 - Prob. 65ECh. 13.4 - Prob. 66ECh. 13.4 - Hypervolume Find the Volume of the...Ch. 13.4 - Prob. 68ECh. 13.5 - Explain how cylindrical coordinates are used to...Ch. 13.5 - Explain how spherical coordinates are used to...Ch. 13.5 - Describe the set {(r, , z): r = 4z} in cylindrical...Ch. 13.5 - Describe the set {(, , ): = /4} in spherical...Ch. 13.5 - Explain why dz r dr d is the volume of a small box...Ch. 13.5 - Explain why 2 sin d d d is the volume of a small...Ch. 13.5 - Prob. 7ECh. 13.5 - Prob. 8ECh. 13.5 - What coordinate system is suggested if the...Ch. 13.5 - What coordinate system is suggested if the...Ch. 13.5 - Sets in cylindrical coordinates Identify and...Ch. 13.5 - Sets in cylindrical coordinates Identify and...Ch. 13.5 - Sets in cylindrical coordinates Identify and...Ch. 13.5 - Sets in cylindrical coordinates Identify and...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Integrals in cylindrical coordinates Evaluate the...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Which weighs more? For 0 r 1, the solid bounded...Ch. 13.5 - Prob. 28ECh. 13.5 - Prob. 29ECh. 13.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 13.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 13.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 13.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 13.5 - Volumes in cylindrical coordinates Use cylindrical...Ch. 13.5 - Prob. 35ECh. 13.5 - Sets in spherical coordinates Identify and sketch...Ch. 13.5 - Sets in spherical coordinates Identify and sketch...Ch. 13.5 - Sets in spherical coordinates Identify and sketch...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Integrals in spherical coordinates Evaluate the...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Volumes in spherical coordinates Use spherical...Ch. 13.5 - Explain why or why not Determine whether the...Ch. 13.5 - Spherical to rectangular Convert the equation 2 =...Ch. 13.5 - Spherical to rectangular Convert the equation 2 =...Ch. 13.5 - Prob. 56ECh. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Mass from density Find the mass of the following...Ch. 13.5 - Changing order of integration If possible, write...Ch. 13.5 - Changing order of integration If possible, write...Ch. 13.5 - Prob. 62ECh. 13.5 - Prob. 63ECh. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Miscellaneous volumes Choose the best coordinate...Ch. 13.5 - Prob. 72ECh. 13.5 - Density distribution A right circular cylinder...Ch. 13.5 - Charge distribution A spherical cloud of electric...Ch. 13.5 - Gravitational field due to spherical shell A point...Ch. 13.5 - Water in a gas tank Before a gasoline-powered...Ch. 13.5 - General volume formulas Use integration to find...Ch. 13.5 - Prob. 78ECh. 13.5 - General volume formulas Use integration to find...Ch. 13.5 - Prob. 80ECh. 13.5 - Intersecting spheres One sphere is centered at the...Ch. 13.6 - Explain how to find the balance point for two...Ch. 13.6 - If a thin 1-m cylindrical rod has a density of =...Ch. 13.6 - Explain how to find the center of mass of a thin...Ch. 13.6 - In the integral for the moment Mx of a thin plate,...Ch. 13.6 - Explain how to find the center of mass of a...Ch. 13.6 - In the integral for the moment Mxz with respect to...Ch. 13.6 - Individual masses on a line Sketch the following...Ch. 13.6 - Individual masses on a line Sketch the following...Ch. 13.6 - One-dimensional objects Find the mass and center...Ch. 13.6 - One-dimensional objects Find the mass and center...Ch. 13.6 - One-dimensional objects Find the mass and center...Ch. 13.6 - One-dimensional objects Find the mass and center...Ch. 13.6 - One-dimensional objects Find the mass and center...Ch. 13.6 - Prob. 14ECh. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Centroid calculations Find the mass and centroid...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Variable-density plates Find the center of mass of...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Center of mass of constant-density solids Find the...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Variable-density solids Find the coordinates of...Ch. 13.6 - Explain why or why not Determine whether the...Ch. 13.6 - Limiting center of mass A thin rod of length L has...Ch. 13.6 - Limiting center of mass A thin rod of length L has...Ch. 13.6 - Prob. 42ECh. 13.6 - Two-dimensional plates Find the mass and center of...Ch. 13.6 - Two-dimensional plates Find the mass and center of...Ch. 13.6 - Centroids Use polar coordinates to find the...Ch. 13.6 - Prob. 46ECh. 13.6 - Centroids Use polar coordinates to find the...Ch. 13.6 - Centroids Use polar coordinates to find the...Ch. 13.6 - Prob. 49ECh. 13.6 - Prob. 50ECh. 13.6 - Prob. 51ECh. 13.6 - Prob. 52ECh. 13.6 - Prob. 53ECh. 13.6 - Prob. 54ECh. 13.6 - Centers of mass for general objects Consider the...Ch. 13.6 - Centers of mass for general objects Consider the...Ch. 13.6 - Prob. 57ECh. 13.6 - Centers of mass for general objects Consider the...Ch. 13.6 - Centers of mass for general objects Consider the...Ch. 13.6 - Geographic vs. population center Geographers...Ch. 13.6 - Center of mass on the edge Consider the thin...Ch. 13.6 - Center of mass on the edge Consider the...Ch. 13.6 - Draining a soda can A cylindrical soda can has a...Ch. 13.6 - Triangle medians A triangular region has a base...Ch. 13.6 - The golden earring A disk of radius r is removed...Ch. 13.7 - Suppose S is the unit square in the first quadrant...Ch. 13.7 - Explain how to compute the Jacobian of the...Ch. 13.7 - Using the transformation T: x = u + v, y = u v,...Ch. 13.7 - Suppose S is the unit cube in the first octant of...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Transforming a square Let S = {(u, v): 0 u l, 0 ...Ch. 13.7 - Images of regions Find the image R in the xy-plane...Ch. 13.7 - Images of regions Find the image R in the xy-plane...Ch. 13.7 - Images of regions Find the image R in the xy-plane...Ch. 13.7 - Images of regions Find the image R in the xy-plane...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13.7 - Solve and compute Jacobians Solve the following...Ch. 13.7 - Solve and compute Jacobians Solve the following...Ch. 13.7 - Solve and compute Jacobians Solve the following...Ch. 13.7 - Solve and compute Jacobians Solve the following...Ch. 13.7 - Double integralstransformation given To evaluate...Ch. 13.7 - Double integralstransformation given To evaluate...Ch. 13.7 - Double integralstransformation given To evaluate...Ch. 13.7 - Double integralstransformation given To evaluate...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Double integralsyour choice of transformation...Ch. 13.7 - Jacobians in three variables Evaluate the...Ch. 13.7 - Prob. 38ECh. 13.7 - Jacobians in three variables Evaluate the...Ch. 13.7 - Jacobians in three variables Evaluate the...Ch. 13.7 - Triple integrals Use a change of variables to...Ch. 13.7 - Triple integrals Use a change of variables to...Ch. 13.7 - Triple integrals Use a change of variables to...Ch. 13.7 - Triple integrals Use a change of variables to...Ch. 13.7 - Explain why or why not Determine whether the...Ch. 13.7 - Prob. 46ECh. 13.7 - Prob. 47ECh. 13.7 - Ellipse problems Let R be the region bounded by...Ch. 13.7 - Ellipse problems Let R be the region bounded by...Ch. 13.7 - Ellipse problems Let R be the region bounded by...Ch. 13.7 - Ellipse problems Let R be the region bounded by...Ch. 13.7 - Ellipse problems Let R be the region bounded by...Ch. 13.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 13.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 13.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 13.7 - Ellipsoid problems Let D be the solid bounded by...Ch. 13.7 - Parabolic coordinates Let T be the transformation...Ch. 13.7 - Shear transformations in 3 The transformation T in...Ch. 13.7 - Linear transformations Consider the linear...Ch. 13.7 - Meaning of the Jacobian The Jacobian is a...Ch. 13.7 - Open and closed boxes Consider the region R...Ch. 13 - Explain why or why not Determine whether the...Ch. 13 - Evaluating integrals Evaluate the following...Ch. 13 - Evaluating integrals Evaluate the following...Ch. 13 - Evaluating integrals Evaluate the following...Ch. 13 - Changing the order of integration Assuming f is...Ch. 13 - Changing the order of integration Assuming f is...Ch. 13 - Changing the order of integration Assuming f is...Ch. 13 - Area of plane regions Use double integrals to...Ch. 13 - Area of plane regions Use double integrals to...Ch. 13 - Area of plane regions Use double integrals to...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Miscellaneous double integrals Choose a convenient...Ch. 13 - Cartesian to polar coordinates Evaluate the...Ch. 13 - Cartesian to polar coordinates Evaluate the...Ch. 13 - Computing areas Sketch the following regions and...Ch. 13 - Computing areas Sketch the following regions and...Ch. 13 - Computing areas Sketch the following regions and...Ch. 13 - Average values 22.Find the average value of...Ch. 13 - Average values 23.Find the average distance from...Ch. 13 - Prob. 24RECh. 13 - Prob. 25RECh. 13 - Changing order of integration Rewrite the...Ch. 13 - Triple integrals Evaluate the following integrals,...Ch. 13 - Triple integrals Evaluate the following integrals,...Ch. 13 - Triple integrals Evaluate the following integrals,...Ch. 13 - Triple integrals Evaluate the following integrals,...Ch. 13 - Triple integrals Evaluate the following integrals,...Ch. 13 - Volumes of solids Find the volume of the following...Ch. 13 - Volumes of solids Find the volume of the following...Ch. 13 - Volumes of solids Find the volume of the following...Ch. 13 - Volumes of solids Find the volume of the following...Ch. 13 - Volumes of solids Find the volume of the following...Ch. 13 - Single to double integral Evaluate...Ch. 13 - Tetrahedron limits Let D be the tetrahedron with...Ch. 13 - A polynomial cube Let D = {(x, y, z): 0 x y2, 0 ...Ch. 13 - Average value 40.Find the average of the square of...Ch. 13 - Average value 41.Find the average x-coordinate of...Ch. 13 - Integrals in cylindrical coordinates Evaluate the...Ch. 13 - Integrals in cylindrical coordinates Evaluate the...Ch. 13 - Volumes in cylindrical coordinates Use integration...Ch. 13 - Volumes in cylindrical coordinates Use integration...Ch. 13 - Integrals in spherical coordinates Evaluate the...Ch. 13 - Integrals in spherical coordinates Evaluate the...Ch. 13 - Volumes in spherical coordinates Use integration...Ch. 13 - Volumes in spherical coordinates Use integration...Ch. 13 - Volumes in spherical coordinates Use integration...Ch. 13 - Center of mass of constant-density plates Find the...Ch. 13 - Center of mass of constant-density plates Find the...Ch. 13 - Center of mass of constant-density plates Find the...Ch. 13 - Center of mass of constant-density plates Find the...Ch. 13 - Center of mass of constant-density solids Find the...Ch. 13 - Center of mass of constant-density solids Find the...Ch. 13 - Prob. 57RECh. 13 - Variable-density solids Find the coordinates of...Ch. 13 - Center of mass for general objects Consider the...Ch. 13 - Prob. 60RECh. 13 - Prob. 61RECh. 13 - Prob. 62RECh. 13 - Volume and weight of a fish tank A spherical fish...Ch. 13 - Prob. 65RECh. 13 - Transforming a square Let S = {(u, v): 0 u 1, 0...Ch. 13 - Prob. 67RECh. 13 - Transforming a square Let S = {(u, v): 0 u 1, 0...Ch. 13 - Prob. 69RECh. 13 - Computing Jacobians Compute the Jacobian J(u, v)...Ch. 13 - Prob. 71RECh. 13 - Prob. 72RECh. 13 - Double integralstransformation given To evaluate...Ch. 13 - Double integralstransformation given To evaluate...Ch. 13 - Double integralstransformation given To evaluate...Ch. 13 - Double integralstransformation given To evaluate...Ch. 13 - Double integrals Evaluate the following integrals...Ch. 13 - Double integrals Evaluate the following integrals...Ch. 13 - Triple integrals Use a change of variables to...Ch. 13 - Triple integrals Use a change of variables to...
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- 3. Consider the sequences of functions f₁: [-π, π] → R, sin(n²x) An(2) n f pointwise as (i) Find a function ƒ : [-T,π] → R such that fn n∞. Further, show that fn →f uniformly on [-π,π] as n → ∞. [20 Marks] (ii) Does the sequence of derivatives f(x) has a pointwise limit on [-7, 7]? Justify your answer. [10 Marks]arrow_forward1. (i) Give the definition of a metric on a set X. [5 Marks] (ii) Let X = {a, b, c} and let a function d : XxX → [0, ∞) be defined as d(a, a) = d(b,b) = d(c, c) 0, d(a, c) = d(c, a) 1, d(a, b) = d(b, a) = 4, d(b, c) = d(c,b) = 2. Decide whether d is a metric on X. Justify your answer. = (iii) Consider a metric space (R, d.), where = [10 Marks] 0 if x = y, d* (x, y) 5 if xy. In the metric space (R, d*), describe: (a) open ball B2(0) of radius 2 centred at 0; (b) closed ball B5(0) of radius 5 centred at 0; (c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] [5 Marks] [5 Marks]arrow_forward(c) sphere S10 (0) of radius 10 centred at 0. [5 Marks] 2. Let C([a, b]) be the metric space of continuous functions on the interval [a, b] with the metric doo (f,g) = max f(x)g(x)|. xЄ[a,b] = 1x. Find: Let f(x) = 1 - x² and g(x): (i) do(f, g) in C'([0, 1]); (ii) do(f,g) in C([−1, 1]). [20 Marks] [20 Marks]arrow_forward
- Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNEarrow_forward17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y ANarrow_forward(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Proarrow_forward
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