Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Chapter 15.R, Problem 8CC
To determine
(a)
To write:
The expressions for the mass.
To determine
(b)
To write:
The expressions for the moments about the coordinate planes.
To determine
(c)
To write:
The expression for the coordinates of the center of mass.
To determine
(d)
To write:
The expression for the moments of inertia about the axes.
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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. 4ECh. 15.1 - Prob. 5ECh. 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. 9ECh. 15.1 - 911 Evaluate the double integral by first...
Ch. 15.1 - Prob. 11ECh. 15.1 - The integral R9y2dA, where R=[0,4][0,2],...Ch. 15.1 - Prob. 13ECh. 15.1 - Prob. 14ECh. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 16ECh. 15.1 - Prob. 17ECh. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 19ECh. 15.1 - Prob. 20ECh. 15.1 - 1526 Calculate the iterated integral....Ch. 15.1 - Prob. 22ECh. 15.1 - Prob. 23ECh. 15.1 - Prob. 24ECh. 15.1 - Prob. 25ECh. 15.1 - Prob. 26ECh. 15.1 - 2734 Calculate the double integral....Ch. 15.1 - Prob. 28ECh. 15.1 - Prob. 29ECh. 15.1 - Prob. 30ECh. 15.1 - Prob. 31ECh. 15.1 - Prob. 32ECh. 15.1 - Prob. 33ECh. 15.1 - Prob. 34ECh. 15.1 - 3536 Sketch the solid whose volume is given by the...Ch. 15.1 - Prob. 36ECh. 15.1 - Prob. 37ECh. 15.1 - Prob. 38ECh. 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. 41ECh. 15.1 - Prob. 42ECh. 15.1 - Prob. 43ECh. 15.1 - Prob. 44ECh. 15.1 - Prob. 45ECh. 15.1 - Prob. 46ECh. 15.1 - 4748 Find the average value of f over the given...Ch. 15.1 - Prob. 48ECh. 15.1 - Prob. 49ECh. 15.1 - Use symmetry to evaluate the double integral....Ch. 15.1 - Prob. 51ECh. 15.1 - a In what way are the theorems of Fubini and...Ch. 15.2 - 16 Evaluate the iterated integral. 150x(8x2y)dydxCh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5ECh. 15.2 - 16 Evaluate the iterated integral. 010ev1+evdwdvCh. 15.2 - Prob. 7ECh. 15.2 - Prob. 8ECh. 15.2 - Prob. 9ECh. 15.2 - Prob. 10ECh. 15.2 - Draw an example of a region that is atype I but...Ch. 15.2 - Prob. 12ECh. 15.2 - Prob. 13ECh. 15.2 - Prob. 14ECh. 15.2 - Prob. 15ECh. 15.2 - 1516 Set up iterated integrals for both orders of...Ch. 15.2 - Prob. 17ECh. 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. 24ECh. 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. 27ECh. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 29ECh. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 31ECh. 15.2 - 2332 Find the volume of the given solid. Bounded...Ch. 15.2 - Prob. 33ECh. 15.2 - Prob. 34ECh. 15.2 - Prob. 35ECh. 15.2 - 3538 Find the volume of the solid by subtracting...Ch. 15.2 - Prob. 37ECh. 15.2 - Prob. 38ECh. 15.2 - Prob. 39ECh. 15.2 - Prob. 40ECh. 15.2 - Prob. 41ECh. 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. 44ECh. 15.2 - Prob. 45ECh. 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. 49ECh. 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. 54ECh. 15.2 - 5156 Evaluate the integral by reversing the order...Ch. 15.2 - Prob. 56ECh. 15.2 - 5758 Express D as a union of regions of type I and...Ch. 15.2 - Prob. 58ECh. 15.2 - 5960 Use Property 11 to estimate the value of the...Ch. 15.2 - Prob. 60ECh. 15.2 - 6162 Find the average value of f over the region...Ch. 15.2 - Prob. 62ECh. 15.2 - Prob. 63ECh. 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. 67ECh. 15.2 - Prob. 68ECh. 15.2 - Prob. 69ECh. 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. 5ECh. 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. 14ECh. 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. 23ECh. 15.3 - Prob. 24ECh. 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. 27ECh. 15.3 - Prob. 28ECh. 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. 32ECh. 15.3 - Prob. 33ECh. 15.3 - Prob. 34ECh. 15.3 - A swimming pool is circular with a 40-ft diameter....Ch. 15.3 - Prob. 36ECh. 15.3 - Prob. 37ECh. 15.3 - Prob. 38ECh. 15.3 - Use polar coordinates to combine the sum...Ch. 15.3 - Prob. 40ECh. 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. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - 310 Find the mass and center of mass of the lamina...Ch. 15.4 - Prob. 7ECh. 15.4 - Prob. 8ECh. 15.4 - Prob. 9ECh. 15.4 - Prob. 10ECh. 15.4 - A lamina occupies the part of the disk x2+y21 in...Ch. 15.4 - Prob. 12ECh. 15.4 - Prob. 13ECh. 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. 16ECh. 15.4 - Prob. 17ECh. 15.4 - Prob. 18ECh. 15.4 - Find the moments of inertia Ix,Iy,I0 for the...Ch. 15.4 - Prob. 20ECh. 15.4 - Prob. 21ECh. 15.4 - Prob. 22ECh. 15.4 - Prob. 23ECh. 15.4 - Prob. 24ECh. 15.4 - Prob. 25ECh. 15.4 - Prob. 26ECh. 15.4 - Prob. 27ECh. 15.4 - Prob. 28ECh. 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. 3ECh. 15.5 - Prob. 4ECh. 15.5 - Prob. 5ECh. 15.5 - 112 Find the area of the surface. The part of the...Ch. 15.5 - Prob. 7ECh. 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. 16ECh. 15.5 - Find the exact area of the surface...Ch. 15.5 - Prob. 18ECh. 15.5 - Find, to four decimal places, the area of the part...Ch. 15.5 - Prob. 20ECh. 15.5 - Show that the area of the part of the plane...Ch. 15.5 - Prob. 22ECh. 15.5 - Find the area of the finite part of the paraboloid...Ch. 15.5 - Prob. 24ECh. 15.6 - Evaluate the integral in Example 1, integrating...Ch. 15.6 - Prob. 2ECh. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - 38 Evaluate the iterated integral....Ch. 15.6 - Prob. 5ECh. 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. 9ECh. 15.6 - 918 Evaluate the triple integral. Eez/ydV, where...Ch. 15.6 - Prob. 11ECh. 15.6 - 918 Evaluate the triple integral. EsinydV, where E...Ch. 15.6 - Prob. 13ECh. 15.6 - 918 Evaluate the triple integral. E(xy)dV, whereE...Ch. 15.6 - Prob. 15ECh. 15.6 - 918 Evaluate the triple integral. TxzdV, where T...Ch. 15.6 - Prob. 17ECh. 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. 21ECh. 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. 26ECh. 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. 30ECh. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 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. 37ECh. 15.6 - Prob. 38ECh. 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. 41ECh. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.6 - 4346 Assume that the solid has constant density k....Ch. 15.6 - Prob. 45ECh. 15.6 - 4346 Assume that the solid has constant density k....Ch. 15.6 - Prob. 47ECh. 15.6 - Prob. 48ECh. 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. 52ECh. 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. 2ECh. 15.7 - Prob. 3ECh. 15.7 - Prob. 4ECh. 15.7 - 56 Describe in words the surface whose equation is...Ch. 15.7 - Prob. 6ECh. 15.7 - 78 Identify the surface whose equation is given....Ch. 15.7 - Prob. 8ECh. 15.7 - Prob. 9ECh. 15.7 - 910 Write the equations in cylindrical...Ch. 15.7 - Prob. 11ECh. 15.7 - 1112 Sketch the solid described by the given...Ch. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Prob. 15ECh. 15.7 - 1516 Sketch the solid whose volume is given by the...Ch. 15.7 - 1728 Use cylindrical coordinates. Evaluate...Ch. 15.7 - Prob. 18ECh. 15.7 - 1728 Use cylindrical coordinates. Evaluate...Ch. 15.7 - Prob. 20ECh. 15.7 - 1728 Use cylindrical coordinates. Evaluate Ex2dV...Ch. 15.7 - Prob. 22ECh. 15.7 - Prob. 23ECh. 15.7 - Prob. 24ECh. 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. 28ECh. 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. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Prob. 4ECh. 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. 7ECh. 15.8 - Prob. 8ECh. 15.8 - Prob. 9ECh. 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. 12ECh. 15.8 - Prob. 13ECh. 15.8 - Prob. 14ECh. 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. 18ECh. 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. 22ECh. 15.8 - Prob. 23ECh. 15.8 - Prob. 24ECh. 15.8 - 2134 Use spherical coordinates. Evaluate...Ch. 15.8 - Prob. 26ECh. 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. 31ECh. 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. 34ECh. 15.8 - 3540 Use cylindrical or spherical coordinates,...Ch. 15.8 - Prob. 36ECh. 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. 48ECh. 15.8 - a Use cylindrical coordinates to show that the...Ch. 15.9 - 16 Find the Jacobian of the transformation....Ch. 15.9 - Prob. 2ECh. 15.9 - Prob. 3ECh. 15.9 - Prob. 4ECh. 15.9 - Prob. 5ECh. 15.9 - 16 Find the Jacobian of the transformation....Ch. 15.9 - Prob. 7ECh. 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. 10ECh. 15.9 - Prob. 11ECh. 15.9 - 1114 A region R in the xy-plane is given. Find...Ch. 15.9 - Prob. 13ECh. 15.9 - 1114 A region R in the xy-plane is given. Find...Ch. 15.9 - Prob. 15ECh. 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. 18ECh. 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. 23ECh. 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. 1CCCh. 15.R - Prob. 2CCCh. 15.R - Prob. 3CCCh. 15.R - Prob. 4CCCh. 15.R - Prob. 5CCCh. 15.R - Prob. 6CCCh. 15.R - a Write the definition of the triple integral of f...Ch. 15.R - Prob. 8CCCh. 15.R - Prob. 9CCCh. 15.R - Prob. 10CCCh. 15.R - Prob. 1TFQCh. 15.R - Prob. 2TFQCh. 15.R - Prob. 3TFQCh. 15.R - Prob. 4TFQCh. 15.R - Prob. 5TFQCh. 15.R - Prob. 6TFQCh. 15.R - Prob. 7TFQCh. 15.R - Prob. 8TFQCh. 15.R - Prob. 9TFQCh. 15.R - A contour map is shown for a function f on the...Ch. 15.R - Prob. 2ECh. 15.R - Prob. 3ECh. 15.R - Prob. 4ECh. 15.R - Prob. 5ECh. 15.R - Prob. 6ECh. 15.R - Prob. 7ECh. 15.R - Prob. 8ECh. 15.R - Prob. 9ECh. 15.R - 910 Write Rf(x,y)dA as an iterated integral, where...Ch. 15.R - Prob. 11ECh. 15.R - Prob. 12ECh. 15.R - Prob. 13ECh. 15.R - Prob. 14ECh. 15.R - Prob. 15ECh. 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. 21ECh. 15.R - Prob. 22ECh. 15.R - Prob. 23ECh. 15.R - Prob. 24ECh. 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. 28ECh. 15.R - Prob. 29ECh. 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. 32ECh. 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. 35ECh. 15.R - 3540 Find the volume of the given solid. Under the...Ch. 15.R - Prob. 37ECh. 15.R - Prob. 38ECh. 15.R - Prob. 39ECh. 15.R - Prob. 40ECh. 15.R - Consider a lamina that occupies the region D...Ch. 15.R - Prob. 42ECh. 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. 45ECh. 15.R - Graph the surface z=xsiny,3x3,y, and find its...Ch. 15.R - Prob. 47ECh. 15.R - Prob. 48ECh. 15.R - Prob. 49ECh. 15.R - Prob. 50ECh. 15.R - Prob. 51ECh. 15.R - Prob. 52ECh. 15.R - Prob. 53ECh. 15.R - Prob. 54ECh. 15.R - Prob. 55ECh. 15.R - Use the transformation x=u2,y=v2,z=w2 to find the...Ch. 15.R - Prob. 57ECh. 15.R - The Mean Value Theorem for double integrals says...Ch. 15.R - Prob. 59ECh. 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. 3PCh. 15.P - If a, b, and c are constant vectors, r is the...Ch. 15.P - Prob. 5PCh. 15.P - Leonhard Euler was able to find the exact sum of...Ch. 15.P - Prob. 7PCh. 15.P - Show that 0arctanxarctanxxdx=2ln by first...Ch. 15.P - Prob. 9PCh. 15.P - Prob. 10PCh. 15.P - Prob. 11PCh. 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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- please all partsarrow_forwardA plate in the form of an isosceles triangle with base 10 ft and altitude 4 ft is submerged vertically in machine oil as shown in Figure 6.8.6a. Find the fluid force F against the plate surface if the oil has weight density ρ = 30 lb/ft³ .arrow_forwardFourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F= -KVT, which means that heat energy flows from hot regions to cold regions. The constant k> 0 is called Fonds=- the conductivity, which has metric units of J/(m-s-K). A temperature function T for a region D is given below. Find the net outward heat flux -KSS VT n dS across the boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k=1. T(x,y,z)=85ex²-y²-2²: D is the sphere of radius a centered at the origin. The net outward heat flux across the boundary is 480x³ (Type an exact answer, using x as needed.)arrow_forward
- Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, F = -KVT, which means that heat energy flows from hot regions to cold regions. The constant k is called the conductivity, which has metric units SS S of J/m-s-K or W/m-K. A temperature function T for a region D is given below. Find the net outward heat flux boundary S of D. It may be easier to use the Divergence Theorem and evaluate a triple integral. Assume that k = 1. T(x,y,z) = 100 - 5x+ 5y +z; D = {(x,y,z): 0≤x≤5, 0≤y≤4, 0≤z≤ 1} The net outward heat flux across the boundary is (Type an exact answer, using as needed.) -KSS S F.ndS = -k VT n dS across thearrow_forwardFind an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 – cos (v)) sin (u), z = sin (v) at the image of a point (u, v) for –x < v < a and -a < u < a. (Write your solution using the form (*,*,*). Use symbolic notation and fractions where needed.) no = Identify this surface. ellipsoid hyperboloid torus cylinderarrow_forwardCan This question be answered with detailed explinations to help better understand the concept.arrow_forward
- Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 5x² - 3xy + xyz (a) Find the rate of change of the potential at P(5, 2, 5) in the direction of the vector v = i + j - k. (b) In which direction does V change most rapidly at P? 73°F Mostly cloudy (c) What is the maximum rate of change at P? Need Help? Read It Show My Work (Optional)? Watch It Qarrow_forwardFind Components of the Acceleration For the curve defined by 7(t) = (e-t cos(t), eʻ sin(t)) find the unit tangent vector, unit normal vector, normal acceleration, and tangential acceleration at t = 3 T(플) - N(플) - 3 3 AN =arrow_forward(i) Using Laplace's equation, find v,and və for inviscid vortex flow. (ii) A rough rule of thumb is that the radius of the eye of a typhoon is 30 m. What is the pressure in the eye of a typhoon with a maximum velocity of 50 m/s, assuming normal atmospheric pressure far afield? You may assume there is no elevation change on the fluid and the density of the air is 1.23 kg/m³.arrow_forward
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