1 Functions 2 Limits And Continuity 3 Derivatives 4 Application Of Derivatives 5 Integrals 6 Applications Of Definite Integrals 7 Trascendental Functions 8 Techniques Of Integration 9 First-order Differential Equations 10 Infinite Sequences And Series 11 Parametric Equations And Polar Coordinates 12 Vectors And The Geometry Of Space 13 Vector-valued Functions And Motion In Space 14 Partial Derivatives 15 Multiple Integrals 16 Integrals And Vector Fields 17 Second-order Differential Equations A.1 Real Numbers And The Real Line A.2 Mathematical Induction A.3 Lines, Circles, And Parabolas A.4 Proofs Of Limit Theorems A.5 Commonly Occurring Limits A.6 Theory Of The Real Numbers A.7 Complex Numbers A.8 The Distributive Law For Vector Cross Products A.9 The Mixed Derivative Theorem And The Increment Theorem expand_more
15.1 Double And Iterated Integrals Over Rectangles 15.2 Double Integrals Over General Regions 15.3 Area By Double Integration 15.4 Double Integrals In Polar Form 15.5 Triple Integrals In Rectangular Coordinates 15.6 Applications 15.7 Triple Integrals In Cylindrical And Spherical Coordinates 15.8 Substitution In Multiple Integrals Chapter Questions expand_more
Problem 1E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 2E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 3E Problem 4E Problem 5E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 6E Problem 7E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 8E Problem 9E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 10E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 11E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 12E: In Exercises 1–12, sketch the region described by the following cylindrical coordinates in... Problem 13E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 14E Problem 15E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 16E Problem 17E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 18E Problem 19E Problem 20E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 21E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 22E: In Exercises 13−22, sketch the region described by the following spherical coordinates in... Problem 23E: Evaluate the cylindrical coordinate integrals in Exercises 23−28.
23.
Problem 24E: Evaluate the cylindrical coordinate integrals in Exercises 23−28.
24.
Problem 25E: Evaluate the cylindrical coordinate integrals in Exercises 23−28.
25.
Problem 26E: Evaluate the cylindrical coordinate integrals in Exercises 23–28.
26.
Problem 27E: Evaluate the cylindrical coordinate integrals in Exercises 23–28.
27.
Problem 28E: Evaluate the cylindrical coordinate integrals in Exercises 23−28.
28.
Problem 29E: The integrals we have seen so far suggest that there are preferred orders of integration for... Problem 30E: The integrals we have seen so far suggest that there are preferred orders of integration for... Problem 31E: The integrals we have seen so far suggest that there are preferred orders of integration for... Problem 32E: The integrals we have seen so far suggest that there are preferred orders of integration for... Problem 33E: Let D be the region bounded below by the plane z = 0, above by the sphere x2 + y2 + z2 = 4, and on... Problem 34E: Let D be the region bounded below by the cone and above by the paraboloid . Set up the triple... Problem 35E: Give the limits of integration for evaluating the integral
as an iterated integral over the region... Problem 36E: Convert the integral
to an equivalent integral in cylindrical coordinates and evaluate the result.
Problem 37E: In Exercises 37–42, set up the iterated integral for evaluating over the given region D.
37. D is... Problem 38E: In Exercises 37–42, set up the iterated integral for evaluating over the given region D.
38. D is... Problem 39E: In Exercises 37–42, set up the iterated integral for evaluating over the given region D.
39. D is... Problem 40E: In Exercises 37–42, set up the iterated integral for evaluating over the given region D.
40. D is... Problem 41E Problem 42E: In Exercises 37–42, set up the iterated integral for evaluating over the given region D.
42. D is... Problem 43E: Evaluate the spherical coordinate integrals in Exercises 43–48.
43.
Problem 44E: Evaluate the spherical coordinate integrals in Exercises 43–48.
44.
Problem 45E: Evaluate the spherical coordinate integrals in Exercises 43–48.
45.
Problem 46E: Evaluate the spherical coordinate integrals in Exercises 43–48.
46.
Problem 47E Problem 48E Problem 49E Problem 50E: The previous integrals suggest there are preferred orders of integration for spherical coordinates,... Problem 51E: The previous integrals suggest there are preferred orders of integration for spherical coordinates,... Problem 52E: The previous integrals suggest there are preferred orders of integration for spherical coordinates,... Problem 53E: Let D be the region in Exercise 33. Set up the triple integrals in spherical coordinates that give... Problem 54E: Let D be the region bounded below by the cone and above by the plane z = 1. Set up the triple... Problem 55E: In Exercises 55–60, (a) find the spherical coordinate limits for the integral that calculates the... Problem 56E: In Exercises 55–60, (a) find the spherical coordinate limits for the integral that calculates the... Problem 57E: In Exercises 55–60, (a) find the spherical coordinate limits for the integral that calculates the... Problem 58E Problem 59E: In Exercises 55–60, (a) find the spherical coordinate limits for the integral that calculates the... Problem 60E: In Exercises 55–60, (a) find the spherical coordinate limits for the integral that calculates the... Problem 61E Problem 62E: Let D be the region in the first octant that is bounded below by the cone and above by the sphere .... Problem 63E: Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center... Problem 64E: Let D be the solid hemisphere x2 + y2 + z2 ≤ 1, z 0. If the density is δ(x, y, z) = 1, express the... Problem 65E: Find the volumes of the solids in Exercises 65–70.
Problem 66E Problem 67E Problem 68E: Find the volumes of the solids in Exercises 65–70.
Problem 69E: Find the volumes of the solids in Exercises 65–70.
69.
Problem 70E: Find the volumes of the solids in Exercises 65–70.
70.
Problem 71E: Sphere and cones Find the volume of the portion of the solid sphere that lies between the cones ... Problem 72E Problem 73E Problem 74E: Cone and planes Find the volume of the solid enclosed by the cone between the planes z = 1 and z =... Problem 75E: Cylinder and paraboloid Find the volume of the region bounded below by the plane z = 0, laterally by... Problem 76E: Cylinder and paraboloids Find the volume of the region bounded below by the paraboloid z = x2 + y2,... Problem 77E: Cylinder and cones Find the volume of the solid cut from the thick-walled cylinder 1 ≤ x2 + y2 ≤ 2... Problem 78E: Sphere and cylinder Find the volume of the region that lies inside the sphere x2 + y2 + z2 = 2 and... Problem 79E Problem 80E: Cylinder and planes Find the volume of the region enclosed by the cylinder x2 + y2 = 4 and the... Problem 81E: Region trapped by paraboloids Find the volume of the region bounded above by the paraboloid z = 5 –... Problem 82E Problem 83E Problem 84E: Sphere and paraboloid Find the volume of the region bounded above by the sphere x2 + y2 + z2 = 2 and... Problem 85E Problem 86E Problem 87E Problem 88E: Find the average value of the function f(ρ, ϕ, θ) = ρ cos ϕ over the solid upper ball ρ ≤ 1, 0 ≤ ϕ ≤... Problem 89E Problem 90E Problem 91E Problem 92E Problem 93E Problem 94E: Centroid Find the centroid of the region cut from the solid ball r2 + z2 ≤ 1 by the half-planes θ =... Problem 95E Problem 96E Problem 97E Problem 98E Problem 99E Problem 100E Problem 101E Problem 102E Problem 103E: Density of center of a planet A planet is in the shape of a sphere of radius R and total mass M with... Problem 104E Problem 105E Problem 106E Problem 107E Problem 108E format_list_bulleted