1 Functions 2 Limits 3 Derivatives 4 Applications Of The Derivative 5 Integration 6 Applications Of Integration 7 Logarithmic And Exponential, And Hyperbolic Functions 8 Integration Techniques 9 Differential Equations 10 Sequences And Infinite Series 11 Power Series 12 Parametric And Polar Curves 13 Vectors And The Geometry Of Space 14 Vector-valued Functions 15 Functions Of Several Variables 16 Multiple Integration 17 Vector Calculus A Proofs Of Selected Theorems B Algebra Review C Complex Numbers expand_more
16.1 Double Integrals Over Rectangular Regions 16.2 Double Integrals Over General Regions 16.3 Double Integrals In Polar Coordinates 16.4 Triple Integrals 16.5 Triple Integrals In Cylindrical And Spherical Coordinates 16.6 Integrals For Mass Calculations 16.7 Change Of Variables In Multiple Integrals Chapter Questions expand_more
Problem 1QC: Describe in polar coordinates the region in the first quadrant between the circles of radius 1 and... Problem 2QC: Express the functions f(x, y) = (x2 + y2)5/2 and h(x, y) = x2 y2 in polar coordinates. Problem 3QC: Give a geometric explanation for the extraneous root z = 4 found in Example 2.
Example 2 Region... Problem 4QC: Express the area of the disk R ={(r, ) : 0 r a, 0 27} in terms of a double integral in polar... Problem 1E: Draw the region {(r, ): 1 r 2, 0 /2}. Why is it called a polar rectangle? Problem 2E: Write the double integral Rf(x,y)dAas an iterated integral in polar coordinates when R = {(r, ): a ... Problem 3E: Sketch in the xy-plane the region of integration for the integral
Problem 4E Problem 5E: How do you find the area of a region R = {(r, ): 0 g( ) r h(), }? Problem 6E: How do you find the average value of a function over a region that is expressed in polar... Problem 7E: Polar rectangles Sketch the following polar rectangles. 7.R = {(r, ): 0 r 5, 0 /2} Problem 8E: Polar rectangles Sketch the following polar rectangles. 8.R = {(r, ): 2 r 3, /4 5/4} Problem 9E: Polar rectangles Sketch the following polar rectangles. 9.R = {(r, ): 1 r 4, /4 2/3} Problem 10E: Polar rectangles Sketch the following polar rectangles. 10.R = {(r, ): 4 r 5, /3 /2} Problem 11E: Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane.... Problem 12E: Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane.... Problem 13E: Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane.... Problem 14E: Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane.... Problem 15E: Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and... Problem 16E: Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and... Problem 17E: Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and... Problem 18E: Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and... Problem 19E: Solids bounded by hyperboloids Find the volume of the solid below the hyperboloid z=51+x2+y2 and... Problem 20E: Solids bounded by hyperboloids Find the volume of the solid below the hyperboloid z=51+x2+y2 and... Problem 21E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 22E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 23E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 24E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 25E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 26E: Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral... Problem 27E: Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r,... Problem 28E: Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r,... Problem 29E: Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r,... Problem 30E: Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r,... Problem 31E: Regions between surfaces Find the volume of the following solid regions. 71.The solid bounded by the... Problem 32E: Volume between surfaces Find the volume of the following solids. 21.The solid bounded by the... Problem 33E: Volume between surfaces Find the volume of the following solids. 19.The solid bounded by the... Problem 34E: Volume between surfaces Find the volume of the following solids. 20.The solid bounded by the... Problem 35E: Volume between surfaces Find the volume of the following solids. 35. The solid bounded below by the... Problem 36E: Volume between surfaces Find the volume of the following solids. 36. The solid bounded by the... Problem 37E: Volume between surfaces Find the volume of the following solids. 37. The solid bounded by the... Problem 38E: Volume between surfaces Find the volume of the following solids. 38. The solid outside the cylinder... Problem 39E: Volume between surfaces Find the volume of the following solids. 39. The solid outside the cylinder... Problem 40E: Volume between surfaces Find the volume of the following solids. 40. The solid bounded by the cone z... Problem 41E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 42E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 43E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 44E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 45E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 46E: Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated... Problem 47E: Computing areas Use a double integral to find the area of the following regions. 47. The annular... Problem 48E: Computing areas Use a double integral to find the area of the following regions. 48. The region... Problem 49E: Computing areas Use a double integral to find the area of the following regions. 49. The region... Problem 50E: Computing areas Use a double integral to find the area of the following regions. 50. The region... Problem 51E: 47-52. Computing areas Use a double integral to find the area of the following regions.
51. The... Problem 52E: Computing areas Use a double integral to find the area of the following regions. 52. The region... Problem 53E: Average values Find the following average values. 45.The average distance between points of the disk... Problem 54E: Average values Find the following average values. 48.The average value of 1/r2 over the annulus {(r,... Problem 55E: Explain why or why not Determine whether the following statements are true and give an explanation... Problem 56E: Areas of circles Use integration to show that the circles r = 2a cos and r = 2a sin have the same... Problem 57E: Filling bowls with water Which bowl holds more water if it is filled to a depth of 4 units? The... Problem 58E: Equal volumes To what height (above the bottom of the bowl) must the cone and paraboloid bowls of... Problem 59E: Volume of a hyperbolic paraboloid Consider the surface z = x2 y2. a.Find the region in the xy-plane... Problem 60E: Volume of a sphere Use double integrals in polar coordinates to verify that the volume of a sphere... Problem 61E: Volume Find the volume of the solid bounded by the cylinder (x 1)2 + y2 = 1, the plane z = 0, and... Problem 62E: Volume Find the volume of the solid bounded by the paraboloid z = 2x2 + 2y2, the plane z = 0, and... Problem 63E: Miscellaneous integrals Evaluate the following integrals using the method of your choice. A sketch... Problem 64E: Miscellaneous integrals Evaluate the following integrals using the method of your choice. A sketch... Problem 65E: Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r... Problem 66E: Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r... Problem 67E: Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r... Problem 68E: Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r... Problem 69E Problem 70E: Mass from density data The following table gives the density (in units of g/cm2) at selected points... Problem 71E: A mass calculation Suppose the density of a thin plate represented by the region R is (r, ) (in... Problem 72E: Area formula In Section 12.3 it was shown that the area of a region enclosed by the polar curve r =... Problem 73E: Normal distribution An important integral in statistics associated with the normal distribution is... Problem 74E: Existence of integrals For what values of p does the integral RdA(x2+y2)p exist in the following... Problem 75E: Integrals in strips Consider the integral I=RdA(1+x2+y2)2, where R = {(x, y): 0 x 1, 0 y a}.... format_list_bulleted