Explanation 1) Concept: i. Identify the region E = { ( r , θ , z ) | α ≤ θ ≤ β , h 1 θ ≤ r ≤ h 2 θ , u 1 r , θ ≤ z ≤ u 2 ( r , θ ) in cylindrical coordinates. ii. Use formula to evaluate the given integral. For conversion of rectangular to cylindrical coordinates, use r 2 = x 2 + y 2 , tan ⁡ θ = y x , z = z P ( r , θ , z ) represents a point in the cylindrical coordinate system x = r cos ⁡ θ , y = r sin ⁡ θ , z = z 2) Formula: i) Formula: ∭ E d V = ∫ α β ∫ h 1 θ h 2 θ ∫ u 1 r cos ⁡ θ , r sin ⁡ θ u 1 r cos ⁡ θ , r sin ⁡ θ f r cos ⁡ θ , r sin ⁡ θ r d z d r d θ 3) Given: The solid region lies between the paraboloid z = x 2 + y 2 and the sphere x 2 + y 2 + z 2 = 2 . 4) Calculation: Evaluate the integral using the cylindrical coordinates ( r , θ , z ) : ∭ E d V It is given that E is a solid region that lies between the paraboloid z = x 2 + y 2 and the sphere x 2 + y 2 + z 2 = 2 Therefore, by using the concept in cylindrical coordinates, it is bounded below by paraboloid, z = r 2 and above by sphere r 2 + z 2 = 2 that is z = 2 - r 2 . Therefore, r ≤ z ≤ 2 - r 2 Intersection of paraboloid and sphere is z = 2 - r 2 , z = r 2 that is, r 2 = 2 - r 2 r 2 + r 4 - 2 = 0 ⇒ ( r 2 - 1 ) ( r 2 + 2 ) = 0 ⇒ r 2 = 1 ⇒ r = 1 Therefore, the region E in cylindrical coordinates is E = { ( r , θ , z ) | 0 ≤ θ ≤ 2 π , 0 ≤ r ≤ 1 , r 2 ≤ z ≤ 2 - r 2 By using the formula and the given solid region, ∭ E

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 15.7, Problem 24E

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Cylindrical coordinates to find the volume of solid.

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Chapter 15.7, Problem 23E

Chapter 15.7, Problem 25E

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Use cylindrical coordinates. Find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z² = 2.

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Chapter 15 Solutions

Calculus (MindTap Course List)

Ch. 15.1 - a Estimate the volume of the solid that lies below...Ch. 15.1 - If R=[0,4][1,2], use a Riemann sum with m=2, n=3...Ch. 15.1 - a Use a Riemann sum with m=n=2 to estimate the...Ch. 15.1 - Prob. 4E Ch. 15.1 - Prob. 5E Ch. 15.1 - A 20-ft-by-30-ft swimming pool is filled with...Ch. 15.1 - A contour map is shown for a function f on the...Ch. 15.1 - The contour map shows the temperature, in degrees...Ch. 15.1 - Prob. 9E Ch. 15.1 - 911 Evaluate the double integral by first...

Ch. 15.1 - Prob. 11E Ch. 15.1 - The integral R9y2dA, where R=[0,4][0,2],...Ch. 15.1 - Prob. 13E Ch. 15.1 - Prob. 14E Ch. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 16E Ch. 15.1 - Prob. 17E Ch. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 19E Ch. 15.1 - Prob. 20E Ch. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 22E Ch. 15.1 - Prob. 23E Ch. 15.1 - Prob. 24E Ch. 15.1 - Prob. 25E Ch. 15.1 - Prob. 26E Ch. 15.1 - 2734 Calculate the double integral....Ch. 15.1 - Prob. 28E Ch. 15.1 - Prob. 29E Ch. 15.1 - Prob. 30E Ch. 15.1 - Prob. 31E Ch. 15.1 - Prob. 32E Ch. 15.1 - Prob. 33E Ch. 15.1 - Prob. 34E Ch. 15.1 - 3536 Sketch the solid whose volume is given by the...Ch. 15.1 - Prob. 36E Ch. 15.1 - Prob. 37E Ch. 15.1 - Prob. 38E Ch. 15.1 - Find the volume of the solid that lying under the...Ch. 15.1 - Find the volume of the solid enclosed by the...Ch. 15.1 - Prob. 41E Ch. 15.1 - Prob. 42E Ch. 15.1 - Prob. 43E Ch. 15.1 - Prob. 44E Ch. 15.1 - Prob. 45E Ch. 15.1 - Prob. 46E Ch. 15.1 - 4748 Find the average value of f over the given...Ch. 15.1 - Prob. 48E Ch. 15.1 - Prob. 49E Ch. 15.1 - Use symmetry to evaluate the double integral....Ch. 15.1 - Prob. 51E Ch. 15.1 - a In what way are the theorems of Fubini and...Ch. 15.2 - 16 Evaluate the iterated integral. 150x(8x2y)dydx Ch. 15.2 - Prob. 2E Ch. 15.2 - Prob. 3E Ch. 15.2 - Prob. 4E Ch. 15.2 - Prob. 5E Ch. 15.2 - 16 Evaluate the iterated integral. 010ev1+evdwdv Ch. 15.2 - Prob. 7E Ch. 15.2 - Prob. 8E Ch. 15.2 - Prob. 9E Ch. 15.2 - Prob. 10E Ch. 15.2 - Draw an example of a region that is atype I but...Ch. 15.2 - Prob. 12E Ch. 15.2 - Prob. 13E Ch. 15.2 - Prob. 14E Ch. 15.2 - Prob. 15E Ch. 15.2 - 1516 Set up iterated integrals for both orders of...Ch. 15.2 - Prob. 17E Ch. 15.2 - 1722 Evaluate the double integral. D(x2+2y)dA, D...Ch. 15.2 - 1722 Evaluate the double integral. Dy2dA, D is...Ch. 15.2 - 1722 Evaluate the double integral. DxydA, D is...Ch. 15.2 - 1722 Evaluate the double integral. D(2xy)dA, D is...Ch. 15.2 - 1722 Evaluate the double integral. DydA, D is...Ch. 15.2 - 2332 Find the volume of the given solid. Under the...Ch. 15.2 - Prob. 24E Ch. 15.2 - 2332 Find the volume of the given solid. Under the...Ch. 15.2 - 2332 Find the volume of the given solid. Enclosed...Ch. 15.2 - Prob. 27E Ch. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 29E Ch. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 31E Ch. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 33E Ch. 15.2 - Prob. 34E Ch. 15.2 - Prob. 35E Ch. 15.2 - 3538 Find the volume of the solid by subtracting...Ch. 15.2 - Prob. 37E Ch. 15.2 - Prob. 38E Ch. 15.2 - Prob. 39E Ch. 15.2 - Prob. 40E Ch. 15.2 - Prob. 41E Ch. 15.2 - 4144 Use a computer algebra system to find the...Ch. 15.2 - 4144 Use a computer algebra system to find the...Ch. 15.2 - Prob. 44E Ch. 15.2 - Prob. 45E Ch. 15.2 - 4550 Sketch the region of integration and change...Ch. 15.2 - 4550 Sketch the region of integration and change...Ch. 15.2 - 4550 Sketch the region of integration and change...Ch. 15.2 - Prob. 49E Ch. 15.2 - 4550 Sketch the region of integration and change...Ch. 15.2 - 5156 Evaluate the integral by reversing the order...Ch. 15.2 - 5156 Evaluate the integral by reversing the order...Ch. 15.2 - 5156 Evaluate the integral by reversing the order...Ch. 15.2 - Prob. 54E Ch. 15.2 - 5156 Evaluate the integral by reversing the order...Ch. 15.2 - Prob. 56E Ch. 15.2 - 5758 Express D as a union of regions of type I and...Ch. 15.2 - Prob. 58E Ch. 15.2 - 5960 Use Property 11 to estimate the value of the...Ch. 15.2 - Prob. 60E Ch. 15.2 - 6162 Find the average value of f over the region...Ch. 15.2 - Prob. 62E Ch. 15.2 - Prob. 63E Ch. 15.2 - In evaluating a double integral over a region D, a...Ch. 15.2 - 6569 Use geometry or symmetry, or both, to...Ch. 15.2 - 6569 Use geometry or symmetry, or both, to...Ch. 15.2 - Prob. 67E Ch. 15.2 - Prob. 68E Ch. 15.2 - Prob. 69E Ch. 15.2 - Graph the solid bounded by the plane x+y+z=1 and...Ch. 15.3 - 14 A region R is shown. Decide whether to use...Ch. 15.3 - 14 A region R is shown. Decide whether to use...Ch. 15.3 - 14 A region R is shown. Decide whether to use...Ch. 15.3 - 14 A region R is shown. Decide whether to use...Ch. 15.3 - Prob. 5E Ch. 15.3 - 56 Sketch the region whose area is given by the...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - 714 Evaluate the given integral by changing to...Ch. 15.3 - Prob. 14E Ch. 15.3 - 1518 Use a double integral to find the area of the...Ch. 15.3 - 1518 Use a double integral to find the area of the...Ch. 15.3 - 1518 Use a double integral to find the area of the...Ch. 15.3 - 1518 Use a double integral to find the area of the...Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - Prob. 23E Ch. 15.3 - Prob. 24E Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - 1927 Use polar coordinates to find the volume of...Ch. 15.3 - Prob. 27E Ch. 15.3 - Prob. 28E Ch. 15.3 - 2932 Evaluate the iterated integral by converting...Ch. 15.3 - 2932 Evaluate the iterated integral by converting...Ch. 15.3 - 2932 Evaluate the iterated integral by converting...Ch. 15.3 - Prob. 32E Ch. 15.3 - Prob. 33E Ch. 15.3 - Prob. 34E Ch. 15.3 - A swimming pool is circular with a 40-ft diameter....Ch. 15.3 - Prob. 36E Ch. 15.3 - Prob. 37E Ch. 15.3 - Prob. 38E Ch. 15.3 - Use polar coordinates to combine the sum...Ch. 15.3 - Prob. 40E Ch. 15.3 - Use the result of Exercise 40 partc to evaluate...Ch. 15.4 - Electric charge is distributed over the rectangle...Ch. 15.4 - Electric charge is distributed over the disk...Ch. 15.4 - Prob. 3E Ch. 15.4 - Prob. 4E Ch. 15.4 - Prob. 5E Ch. 15.4 - 310 Find the mass and center of mass of the lamina...Ch. 15.4 - Prob. 7E Ch. 15.4 - Prob. 8E Ch. 15.4 - Prob. 9E Ch. 15.4 - Prob. 10E Ch. 15.4 - A lamina occupies the part of the disk x2+y21 in...Ch. 15.4 - Prob. 12E Ch. 15.4 - Prob. 13E Ch. 15.4 - Find the center of mass of the lamina in Exercise...Ch. 15.4 - Find the center of mass of a lamina in the shape...Ch. 15.4 - Prob. 16E Ch. 15.4 - Prob. 17E Ch. 15.4 - Prob. 18E Ch. 15.4 - Find the moments of inertia Ix,Iy,I0 for the...Ch. 15.4 - Prob. 20E Ch. 15.4 - Prob. 21E Ch. 15.4 - Prob. 22E Ch. 15.4 - Prob. 23E Ch. 15.4 - Prob. 24E Ch. 15.4 - Prob. 25E Ch. 15.4 - Prob. 26E Ch. 15.4 - Prob. 27E Ch. 15.4 - Prob. 28E Ch. 15.4 - Suppose X and Y are random variables with joint...Ch. 15.4 - a A lamp has two bulbs, each of a type with...Ch. 15.4 - Suppose that X and Y are independent random...Ch. 15.4 - Xavier and Yolanda both have classes that end at...Ch. 15.4 - When studying the spread of an epidemic, we assume...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - Prob. 3E Ch. 15.5 - Prob. 4E Ch. 15.5 - Prob. 5E Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - Prob. 7E Ch. 15.5 - 112 Find the area of the surface. The surface...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - 1314 Find the area of the surface correct to four...Ch. 15.5 - 1314 Find the area of the surface correct to four...Ch. 15.5 - a Use the Midpoint Rule for double integrals see...Ch. 15.5 - Prob. 16E Ch. 15.5 - Find the exact area of the surface...Ch. 15.5 - Prob. 18E Ch. 15.5 - Find, to four decimal places, the area of the part...Ch. 15.5 - Prob. 20E Ch. 15.5 - Show that the area of the part of the plane...Ch. 15.5 - Prob. 22E Ch. 15.5 - Find the area of the finite part of the paraboloid...Ch. 15.5 - Prob. 24E Ch. 15.6 - Evaluate the integral in Example 1, integrating...Ch. 15.6 - Prob. 2E Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - Prob. 5E Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - Prob. 9E Ch. 15.6 - 918 Evaluate the triple integral. Eez/ydV, where...Ch. 15.6 - Prob. 11E Ch. 15.6 - 918 Evaluate the triple integral. EsinydV, where E...Ch. 15.6 - Prob. 13E Ch. 15.6 - 918 Evaluate the triple integral. E(xy)dV, whereE...Ch. 15.6 - Prob. 15E Ch. 15.6 - 918 Evaluate the triple integral. TxzdV, where T...Ch. 15.6 - Prob. 17E Ch. 15.6 - 918 Evaluate the triple integral. EzdV, where E is...Ch. 15.6 - 1922 Use a triple integral to find the volume of...Ch. 15.6 - 1922 Use a triple integral to find the volume of...Ch. 15.6 - Prob. 21E Ch. 15.6 - 1922 Use a triple integral to find the volume of...Ch. 15.6 - a Express the volume of the wedge in the first...Ch. 15.6 - a In the Midpoint Rule for triple integrals we use...Ch. 15.6 - 2526 Use the Midpoint Rule for triple integrals...Ch. 15.6 - Prob. 26E Ch. 15.6 - 2728 Sketch the solid whose volume is given by the...Ch. 15.6 - 2728 Sketch the solid whose volume is given by the...Ch. 15.6 - 2932 Express the integral Ef(x,y,z)dV as an...Ch. 15.6 - Prob. 30E Ch. 15.6 - Prob. 31E Ch. 15.6 - Prob. 32E Ch. 15.6 - The figure shows the region of integration for the...Ch. 15.6 - The figure shows the region of integration for the...Ch. 15.6 - 3536 Write five other iterated integrals that are...Ch. 15.6 - 3536 Write five other iterated integrals that are...Ch. 15.6 - Prob. 37E Ch. 15.6 - Prob. 38E Ch. 15.6 - 3942 Find the mass and center of mass of the solid...Ch. 15.6 - 3942 Find the mass and center of mass of the solid...Ch. 15.6 - Prob. 41E Ch. 15.6 - Prob. 42E Ch. 15.6 - Prob. 43E Ch. 15.6 - 4346 Assume that the solid has constant density k....Ch. 15.6 - Prob. 45E Ch. 15.6 - 4346 Assume that the solid has constant density k....Ch. 15.6 - Prob. 47E Ch. 15.6 - Prob. 48E Ch. 15.6 - Let E be the solid in the first octant bounded by...Ch. 15.6 - If E is the solid of Exercise 18 with density...Ch. 15.6 - The joint density function for random variables X,...Ch. 15.6 - Prob. 52E Ch. 15.6 - 5354 The average value of a function f(x,y,z)over...Ch. 15.6 - 5354 The average value of a function f(x,y,z)over...Ch. 15.6 - a Find the region E for which the triple integral...Ch. 15.7 - 12 Plot the point whose cylindrical coordinates...Ch. 15.7 - Prob. 2E Ch. 15.7 - Prob. 3E Ch. 15.7 - Prob. 4E Ch. 15.7 - 56 Describe in words the surface whose equation is...Ch. 15.7 - Prob. 6E Ch. 15.7 - 78 Identify the surface whose equation is given....Ch. 15.7 - Prob. 8E Ch. 15.7 - Prob. 9E Ch. 15.7 - 910 Write the equations in cylindrical...Ch. 15.7 - Prob. 11E Ch. 15.7 - 1112 Sketch the solid described by the given...Ch. 15.7 - Prob. 13E Ch. 15.7 - Prob. 14E Ch. 15.7 - Prob. 15E Ch. 15.7 - 1516 Sketch the solid whose volume is given by the...Ch. 15.7 - 1728 Use cylindrical coordinates. Evaluate...Ch. 15.7 - Prob. 18E Ch. 15.7 - 1728 Use cylindrical coordinates. Evaluate...Ch. 15.7 - Prob. 20E Ch. 15.7 - 1728 Use cylindrical coordinates. Evaluate Ex2dV...Ch. 15.7 - Prob. 22E Ch. 15.7 - Prob. 23E Ch. 15.7 - Prob. 24E Ch. 15.7 - 1728 Use cylindrical coordinates. a Find the...Ch. 15.7 - 1728 Use cylindrical coordinates. a Find the...Ch. 15.7 - 1728 Use cylindrical coordinates. Find the mass...Ch. 15.7 - Prob. 28E Ch. 15.7 - 2930 Evaluate the integral by changing to...Ch. 15.7 - 2930 Evaluate the integral by changing to...Ch. 15.7 - When studying the formation of mountain ranges,...Ch. 15.8 - 12 Plot the point whose spherical coordinates are...Ch. 15.8 - Prob. 2E Ch. 15.8 - Prob. 3E Ch. 15.8 - Prob. 4E Ch. 15.8 - 56 Describe in words the surface whose equation is...Ch. 15.8 - 56 Describe in words the surface whose equation is...Ch. 15.8 - Prob. 7E Ch. 15.8 - Prob. 8E Ch. 15.8 - Prob. 9E Ch. 15.8 - 910 Write the equation in spherical coordinates. a...Ch. 15.8 - 1114 Sketch the solid described by the given...Ch. 15.8 - Prob. 12E Ch. 15.8 - Prob. 13E Ch. 15.8 - Prob. 14E Ch. 15.8 - A solid lies above the cone z=x2+y2 and below the...Ch. 15.8 - a Find inequalities that describe a hollow ball...Ch. 15.8 - 1718 Sketch the solid whose volume is given by the...Ch. 15.8 - Prob. 18E Ch. 15.8 - 1920 Set up the triple integral of an arbitrary...Ch. 15.8 - 1920 Set up the triple integral of an arbitrary...Ch. 15.8 - 2134 Use spherical coordinates. Evaluate...Ch. 15.8 - Prob. 22E Ch. 15.8 - Prob. 23E Ch. 15.8 - Prob. 24E Ch. 15.8 - 2134 Use spherical coordinates. Evaluate...Ch. 15.8 - Prob. 26E Ch. 15.8 - 2134 Use spherical coordinates. Find the volume of...Ch. 15.8 - 2134 Use spherical coordinates. Find the average...Ch. 15.8 - 2134 Use spherical coordinates. a Find the volume...Ch. 15.8 - 2134 Use spherical coordinates. Find the volume of...Ch. 15.8 - Prob. 31E Ch. 15.8 - 2134 Use spherical coordinates. Let H be a solid...Ch. 15.8 - 2134 Use spherical coordinates. a Find the...Ch. 15.8 - Prob. 34E Ch. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - Prob. 36E Ch. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - 4143 Evaluate the integral by changing to...Ch. 15.8 - 4143 Evaluate the integral by changing to...Ch. 15.8 - 4143 Evaluate the integral by changing to...Ch. 15.8 - A model for the density of the earths atmosphere...Ch. 15.8 - Use a graphing device to draw a silo consisting of...Ch. 15.8 - The latitude and longitude of a point P in the...Ch. 15.8 - The surfaces =1+15sinmsinn have been used as...Ch. 15.8 - Prob. 48E Ch. 15.8 - a Use cylindrical coordinates to show that the...Ch. 15.9 - 16 Find the Jacobian of the transformation....Ch. 15.9 - Prob. 2E Ch. 15.9 - Prob. 3E Ch. 15.9 - Prob. 4E Ch. 15.9 - Prob. 5E Ch. 15.9 - 16 Find the Jacobian of the transformation....Ch. 15.9 - Prob. 7E Ch. 15.9 - 710 Find the image of the set S under the given...Ch. 15.9 - 710 Find the image of the set S under the given...Ch. 15.9 - Prob. 10E Ch. 15.9 - Prob. 11E Ch. 15.9 - 1114 A region R in the xy-plane is given. Find...Ch. 15.9 - Prob. 13E Ch. 15.9 - 1114 A region R in the xy-plane is given. Find...Ch. 15.9 - Prob. 15E Ch. 15.9 - 1520 Use the given transformation to evaluate the...Ch. 15.9 - 1520 Use the given transformation to evaluate the...Ch. 15.9 - Prob. 18E Ch. 15.9 - 1520 Use the given transformation to evaluate the...Ch. 15.9 - 1520 Use the given transformation to evaluate the...Ch. 15.9 - a Evaluate EdV, where E is the solid enclosed by...Ch. 15.9 - An important problem in thermodynamics is to find...Ch. 15.9 - Prob. 23E Ch. 15.9 - 2327 Evaluate the integral by making an...Ch. 15.9 - 2327 Evaluate the integral by making an...Ch. 15.9 - 2327 Evaluate the integral by making an...Ch. 15.9 - 2327 Evaluate the integral by making an...Ch. 15.9 - Let f be continuous on [0,1] and let R be the...Ch. 15.R - Prob. 1CC Ch. 15.R - Prob. 2CC Ch. 15.R - Prob. 3CC Ch. 15.R - Prob. 4CC Ch. 15.R - Prob. 5CC Ch. 15.R - Prob. 6CC Ch. 15.R - a Write the definition of the triple integral of f...Ch. 15.R - Prob. 8CC Ch. 15.R - Prob. 9CC Ch. 15.R - Prob. 10CC Ch. 15.R - Prob. 1TFQ Ch. 15.R - Prob. 2TFQ Ch. 15.R - Prob. 3TFQ Ch. 15.R - Prob. 4TFQ Ch. 15.R - Prob. 5TFQ Ch. 15.R - Prob. 6TFQ Ch. 15.R - Prob. 7TFQ Ch. 15.R - Prob. 8TFQ Ch. 15.R - Prob. 9TFQ Ch. 15.R - A contour map is shown for a function f on the...Ch. 15.R - Prob. 2E Ch. 15.R - Prob. 3E Ch. 15.R - Prob. 4E Ch. 15.R - Prob. 5E Ch. 15.R - Prob. 6E Ch. 15.R - Prob. 7E Ch. 15.R - Prob. 8E Ch. 15.R - Prob. 9E Ch. 15.R - 910 Write Rf(x,y)dA as an iterated integral, where...Ch. 15.R - Prob. 11E Ch. 15.R - Prob. 12E Ch. 15.R - Prob. 13E Ch. 15.R - Prob. 14E Ch. 15.R - Prob. 15E Ch. 15.R - Sketch the solid consisting of all points with...Ch. 15.R - Describe the region whose area is given by the...Ch. 15.R - Describe the solid whose volume is given by the...Ch. 15.R - 1920 Calculate the iterated integral by first...Ch. 15.R - 1920 Calculate the iterated integral by first...Ch. 15.R - Prob. 21E Ch. 15.R - Prob. 22E Ch. 15.R - Prob. 23E Ch. 15.R - Prob. 24E Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - Prob. 28E Ch. 15.R - Prob. 29E Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - Prob. 32E Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - 2134 Calculate the value of the multiple integral....Ch. 15.R - Prob. 35E Ch. 15.R - 3540 Find the volume of the given solid. Under the...Ch. 15.R - Prob. 37E Ch. 15.R - Prob. 38E Ch. 15.R - Prob. 39E Ch. 15.R - Prob. 40E Ch. 15.R - Consider a lamina that occupies the region D...Ch. 15.R - Prob. 42E Ch. 15.R - a Find the centroid of a solid right circular cone...Ch. 15.R - Find the area of the part of the cone z2=a2(x2+y2)...Ch. 15.R - Prob. 45E Ch. 15.R - Graph the surface z=xsiny,3x3,y, and find its...Ch. 15.R - Prob. 47E Ch. 15.R - Prob. 48E Ch. 15.R - Prob. 49E Ch. 15.R - Prob. 50E Ch. 15.R - Prob. 51E Ch. 15.R - Prob. 52E Ch. 15.R - Prob. 53E Ch. 15.R - Prob. 54E Ch. 15.R - Prob. 55E Ch. 15.R - Use the transformation x=u2,y=v2,z=w2 to find the...Ch. 15.R - Prob. 57E Ch. 15.R - The Mean Value Theorem for double integrals says...Ch. 15.R - Prob. 59E Ch. 15.R - a Evaluate D1(x2+y2)n/2dA, where n is an integer...Ch. 15.P - If x denotes the greatest integer in x, evaluate...Ch. 15.P - Evaluate the integral 0101emax{x2,y2}dydx where...Ch. 15.P - Prob. 3P Ch. 15.P - If a, b, and c are constant vectors, r is the...Ch. 15.P - Prob. 5P Ch. 15.P - Leonhard Euler was able to find the exact sum of...Ch. 15.P - Prob. 7P Ch. 15.P - Show that 0arctanxarctanxxdx=2ln by first...Ch. 15.P - Prob. 9P Ch. 15.P - Prob. 10P Ch. 15.P - Prob. 11P Ch. 15.P - Evaluate limnn2i=1nj=1n21n2+ni+j.Ch. 15.P - The plane xa+yb+zc=1a0,b0,c0 cuts the ellipsoid...

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To use: Cylindrical coordinates to find the volume of solid.

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To use: Cylindrical coordinates to find the volume of solid.

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Chapter 15 Solutions

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Cylindrical coordinates to find the volume of solid.