Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
HOW DO YOU SEE IT? Use each order of integration to write an iterated integral that represents the area of the region R (see figure). (a) Area=dxdy (b) Area=dydx71E72E73E74E75EEvaluating an Iterated Integral Using Technology In Exercises 71-76, use a computer algebra system to evaluate the iterated integral. 0/201+sin15rdrd77EComparing Different Orders of Integration Using Technology In Exercises 77 and 78, (a) sketch the region of integration, (b) change the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. 024x24(x2/4)xyx2+y2+1dydx79E80ECONCEPT CHECK Approximating the Volume of a Solid In your own words, describe the process of using an inner partition to approximate the volume of a solid region lying above the xy-plane. How can the approximation be improved?2EApproximation In Exercises 3-6, approximate the integral Rf(x,y)dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum i=18f(xi,yi)Ai, where (xi,yi) the center of the ith square. Evaluate the iterated integral and compare it with the approximation. 0402(x+y)dydxApproximation In Exercises 3-6, approximate the integral Rf(x,y)dA by dividing the rectangle R with vertices (0, 0), (4, 0), (4, 2), and (0, 2) into eight equal squares and finding the sum i=18f(xi,yi)Ai, where (xi,yi) the center of the ith square. Evaluate the iterated integral and compare it with the approximation. 120402x2ydydx5E6E7E8E9EEvaluating a Double IntegralIn Exercises 712, sketch the region R and evaluate the iterated integral Rf(x,y)dA. 04y/2yx2y2dxdy11EEvaluating a Double Integral In Exercises 712, sketch the region R and evaluate the iterated integral Rf(x,y)dA. 01y10ex+ydxdy+0101yex+ydxdyEvaluating a Double Integral In Exercises 1320, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. RxydAEvaluating a Double IntegralIn Exercises 1320, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. RsinxsinydA R: rectangle with vertices (,0),(,0),(,/2),(,/2)Evaluating a Double IntegralIn Exercises 1320, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. Ryx2+y2dA R: trapezoid bounded by y=x,y=2x,x=1,x=2Evaluating a Double IntegralIn Exercises 1320, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. RxeydA R: triangle bounded by y=4x,y=0,x=017E18EEvaluating a Double IntegralIn Exercises 1320, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. RxdA R: sector of a circle in the first quadrant bounded by y=25x2,3x4y=0,y=020E21EFinding Volume In Exercises 21-26, use a double integral to find the volume of the indicated solid.23EFinding Volume In Exercises 21-26, use a double integral to find the volume of the indicated solid.25EFinding Volume In Exercises 21-26, use a double integral to find the volume of the indicated solid.27E28EFinding Volume In Exercises 29-34, set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations. zxy,z=0,y=x3,x=1,firstoctant30E31EFinding Volume In Exercises 29-34, set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations. z=11+y2,x=0,x=2,y033EFinding Volume In Exercises 29-34, set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations. x2+z2=1,y2+z2=1,firstoctant35E36E37E38E39EVolume of a Region Bounded by Two Surfaces In Exercises 35-40, set up a double integral to find the volume of the solid region bounded by the graphs of the equations. Do not evaluate the integral. z=x2+y2,z=18x2y241E42E43E44E45E46E47E48E49E50E51E52E53E54EAverage Value In Exercises 51-56. find the average value of f(x, y) over the plane region R. f(x,y)=ex+y, R: triangle with vertices (0, 0), (0, 1), (1, 1)56E57E58E59E60E61E62E63E64E65E66E67E68E69E70EMaximizing a Double Integral Determine the region R in the xy-plane that maximizes the value of R(9x2y2)dAMinimizing a Double Integral Determine the region R in the xy-plane that minimizes the value of R(x2+y24)dA73E74E75E76ECONCEPT CHECK Choosing a Coordinate System In Exercise 1 and 2, the region R for the integral Rf(x,y)dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral.CONCEPT CHECK Choosing a Coordinate SystemIn Exercises 1 and 2, the region R for the integral Rf(x,y)dA is shown. State whether you would use rectangular or polar coordinates to evaluate the integral.3E4EDescribing a Region In Exercises 58, use polar coordinates to describe the region shown.Describing a Region In Exercises 58, use polar coordinates to describe the region shown.7EDescribing a Region In Exercises 58, use polar coordinates to describe the region shown.9EEvaluating a Double Integral in Exercises 9-16, evaluate the double Integral Rf(r,)dA and sketch the region R. 0/20sinr2drd11EEvaluating a Double Integral in Exercises 9-16, evaluate the double Integral Rf(r,)dA and sketch the region R. 0/404r2sincosdrd13E14E15E16EConverting to Polar Coordinates: In Exercises 1726, evaluate the iterated integral by converting to polar coordinates. 0309y2ydxdy18EConverting to Polar Coordinates: In Exercises 1726, evaluate the iterated integral by converting to polar coordinates. 2204x2(x2+y2)dydx20EConverting to Polar Coordinates In Exercises 17-26, evaluate the iterated integral by converting to polar coordinates. 0101x2(x2+y2)3/2dydxConverting to Polar Coordinates: In Exercises 1726, evaluate the iterated integral by converting to polar coordinates. 02y8y2x2+y2dxdy23EConverting to Polar Coordinates: In Exercises 1726, evaluate the iterated integral by converting to polar coordinates. 0404yy2x2dxdy25EConverting to Polar Coordinates: In Exercises 1726, evaluate the iterated integral by converting to polar coordinates. 0606x2sinx2+y2dydx27EConverting to Polar Coordinates: In Exercises 27 and 28, combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. 05220xxydydx+5225025x2xydydx29E30EConverting to Polar Coordinates In Exercises 2932, use polar coordinates to set up and evaluate the double integral Rf(x,y)dA f(x,y)=arctanyxR:x2+y21,x2+y24,0yx32E33E34E35E36E37E38E39E40E41E42E43E44EAreaIn Exercises 4146, use a double integral to find the area of the shaded region.AreaIn Exercises 4146, use a double integral to find the area of the shaded region.47E48EArea: In Exercises 4752, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region. Inside the circle r=3cos and outside the cardioid r=1+cosArea: In Exercises 4752, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region. Inside the cardioid r=1+cos and outside the circle r=3cos51EArea: In Exercises, 4752, sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region. Inside the circle r=2 and outside the cardioid r=22cos53E54E55E56EVolume Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations z=25e(x2+y2)/4,z=0, and x2+y2=16 when one-tenth of the volume of the solid is removed.58E59E60E61ETrue or False? In Exercises 61 and 62, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(r,) is a constant function and the area of the region S is twice that of the region R, then 2Rf(r,)dA=sf(r,)dA63E64E65E66E67EArea Show that the area of the polar sector R (see figure) is A=rr, where r=(r1+r2)2 is the average radius of RMass of a Planar Lamina Explain when you should use a double integral to find the mass of a planar lamina.Moment of InertiaDescribe what the moment of inertia measures.Finding the Mass of a Lamina In Exercises 3-6, find the mass of the lamina described by the inequalities, given that its density is (x,y)=xy. 0x2,0y2Finding the Mass of a Lamina In Exercises 3-6, find the mass of the lamina described by the inequalities, given that its density is (x,y)=xy. 0x2,0y4x2Finding the Mass of a Lamina In Exercises 3-6, find the mass of the lamina described by the inequalities, given that its density is (x,y)=xy. 0x1,0y1x26E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24EFinding the Center of Mass Using Technology In Exercises 2528, use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y=ex,y=0,x=0,x=2,=kxy26E27E28E29E30E31E32EFinding the Radius of Gyration About Each Axis in Exercises 29-34, verify the given moment(s) of inertia and find x and y. Assume that each lamina has a density of =1 gramper square centimeter. (These regions are common shapes use din engineering.) Quarter circle34E35E36E37EFinding Moments of Inertia and Radii of Gyration In Exercises 3538, find Ix,Iy,I0,x and y for the lamina bounded by the graphs of the equations. y=x2,y2=x,=kx39E40E41E42E43E44E45E46E47EHOW DO YOU SEE IT? The center of mass of the lamina of constant density shown in the figure is (2, 85 ). Make a conjecture about how the center of mass (x, y) changes for each given nonconstant density (x, y). Explain. (Make your conjecture without performing any calculations.) (a) (x,y)=ky (b) (x,y)=k2x (c) (x,y)=kxy (d) (x,y)=k(4x)(4y)49ECONCEPT CHECK Surface Area What is the differential of surface area dS, in space?CONCEPT CHECK Numerical Integration Write a double integral that represents the surface area of the portion of the plane z=3 that lies above the rectangular region with vertices (0, 0), (4, 0), (0, 5), and (4, 5). Then find the surface area without integrating.Finding Surface AreaIn Exercises 316, find the area of the surface given by z=f(x,y) that lies above the region R. f(x,y)=2x+2y R: triangle with vertices (0,0),(4,0),(0,4)Finding Surface AreaIn Exercises 316, find the area of the surface given by z=f(x,y) that lies above the region R. f(x,y)=15+2x3y R: square with vertices (0,0),(3,0),(0,3),(3,3)Finding Surface Area In Exercises 3-16, find the area of the surface given by z=f(x,y) that lies above the region R. f(x,y)=4+5x+6y,R=(x,y):x2+y246EFinding Surface AreaIn Exercises 316, find the area of the surface given by z=f(x,y) that lies above the region R. f(x,y)=9x2 R: square with vertices (0,0),(2,0),(0,2),(2,2)8E9E10E11E12E13E14E15E16EFinding Surface Area In Exercises 17-20, find the area of the surface. The portion of the plane z=123x2y in the first octantFinding Surface Area In Exercises 17-20, find the area of the surface. The portion of the paraboloid z=16x2y2 in the first octantFinding Surface Area In Exercises 17-20, find the area of the surface. The portion of the sphere x2+y2+z2=25 inside the cylinder x2+y2=920E21E22E23E24E25E26E27E28E29E30E31EHOW DO YOU SEE IT? Consider the surface f(x,y)=x2+y2 (see figure) and the surface area f that lies above each region R. Without integrating, order the surface areas from least to greatest. Explain. (a) R:rectanglewithvertices(0,0),(2,0),(2,2),(0,2) (b) R:trianglewithvertices(0,0),(2,0),(2,2) (c) R={(x,y):x2+y24,firstquadrantonly}33E34EProduct DesignA company produces a spherical object of radius 25 centimeters. A hole of radius 4 centimeters is drilled through the center of the object. (a) Find the volume of the object. (b) Find the outer surface area of the object.Modeling Data A company builds a ware house with dimensions 30 feet by 50 feet. The symmetrical shape and selected heights of the roof are shown in the figure. (a) Use the regression capabilities of a graphing utility to find a model of the form z=ay3+by2+cy+d for the roof line. (b) Use the numerical integration capabilities of a graphing utility and the model in part (a) to approximate the volume of storage space in the ware house. (c) Use the numerical integration capabilities of a graphing utility and the model in part (a) to approximate the surface area of the roof. (d) Approximate the are length of the roof line and find the surface area of the roof by multiplying the arc length by the length of the ware house. Compare the results and the integrations with those found in part (c).37E38ECONCEPT CHECK Triple Integrals What does Q=QdV represent?2EEvaluating a Triple Iterated Integral In Exercises 3-10, evaluate the triple iterated integral. 030201(x+y+z)dxdzdyEvaluating a Triple Iterated Integral In Exercises 3-10, evaluate the triple iterated integral. 020112xyz3dxdydz5E6E7E8E9E10EEvaluating a Triple Iterated Integral Using Technology In Exercises 11 and 12, use a computer algebra system to evaluate the triple iterated integral. 039y29y20y2ydzdxdyEvaluating a Triple Iterated Integral Using Technology In Exercises 11 and 12, use a computer algebra system to evaluate the triple iterated integral. 0302(2y/3)062y3zzex2y2dxdzdySetting Up a Triple IntegralIn Exercises 13-18, set up a triple integral for the volume of the solid. Do not evaluate the integral. The solid in the first octant bounded by the coordinate planes and the plane z=7x2ySetting Up a Triple IntegralIn Exercises 13-18, set up a triple integral for the volume of the solid. Do not evaluate the integral. The solid bounded by z=9x2,z=0,y=0 and y=2xSetting Up a Triple IntegralIn Exercises 13-18, set up a triple integral for the volume of the solid. Do not evaluate the integral. The solid bounded by z=6x2y2 and z=0Setting Up a Triple IntegralIn Exercises 13-18, set up a triple integral for the volume of the solid. Do not evaluate the integral. The solid bounded by z=1x2y2 and z=017E18EVolume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations.Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations.Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z=6x2,y=33x,firstoctantVolume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z=9x3,y=x2+2,y=0,z=0,x0Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z=2y,z=4y2,x=0,x=3,y=0Volume In Exercises 19-24, use a triple integral to find the volume of the solid bounded by the graphs of the equations. z=x,y=x+2,y=x2,firstoctantChanging the Order of integration In Exercises 25-30, sketch the solid whose volume is Riven by the iterated integral. Then rewrite the integral using the indicated order of integration. 01100y2dzdydx Rewrite using dy dz dxChanging the Order of integration In Exercises 25-30, sketch the solid whose volume is Riven by the iterated integral. Then rewrite the integral using the indicated order of integration. 11y2101xdzdxdy Rewrite using dy dz dx27E28EChanging the Order of Integration In Exercises 2530, sketch the solid whose volume is given by the iterated integral. Then rewrite the integral using the indicated order of integration. 01y101y2dzdxdy Rewriteusing dzdydxChanging the Order of integration In Exercises 25-30, sketch the solid whose volume is Riven by the iterated integral. Then rewrite the integral using the indicated order of integration. 022x40y24x2dzdydx Rewrite using dy dz dxOrders of Integration In Exercises 31-34, write a triple integral for f(x,y,z)=xyz over the solid region Q for each of the six possible orders of integration. Then evaluate one of the triple integrals. Q=(x,y,z):0x1,0y5x,0z3Orders of Integration In Exercises 31-34, write a triple integral for f(x,y,z)=xyz over the solid region Q for each of the six possible orders of integration. Then evaluate one of the triple integrals. Q=(x,y,z):0x2,x2y4,0z2x33E34E35EOrders of Integration In Exercises 35 and 36, the figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in the live other orders. 030x09x2dzdydx37E38E39ECenter of Mass In Exercises 37-40, find the mass and the indicated coordinate of the center of mass of the solid region Q of density bounded by the graphs of the equations. Find y using (x,y,z)=k Q:xa+yb+zc=1(a,b,c0),x=0,y=0,z=0Center of Mass In Exercises 41 and 42, set up the triple integrals for finding the mass and the center of mass of the solid of density p bounded by the graphs of the equations. Do not evaluate the integrals. x=0,x=b,y=0,y=b,z=0,z=b,(x,y,z)=kxy42EThink About It The center of mass of a solid of constant density is shown in the figure. In Exercises 43-46, make a conjecture about how the center of mass (x,y,z) will change for the nonconstant density (x,y,z). Explain. (Make your conjecture without performing any calculations.) (x,y,z)=kx44EThink About It The center of mass of a solid of constant density is shown in the figure. In Exercises 43-46, make a conjecture about how the center of mass (x,y,z) will change for the nonconstant density (x,y,z). Explain. (Make your conjecture without performing any calculations.) (x,y,z)=k(y+2)Think About It The center of mass of a solid of constant density is shown in the figure. In Exercises 43-46, make a conjecture about how the center of mass (x,y,z) will change for the nonconstant density (x,y,z). Explain. (Make your conjecture without performing any calculations.) (x,y,z)=kxz2(y+2)2Centroid In Exercises 47-52, find the centroid of the solid region hounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) z=hrx2+y2,z=hCentroid In Exercises 47-52, find the centroid of the solid region hounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) y=9x2,z=y,z=049ECentroid In Exercises 47-52, find the centroid of the solid region hounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) z=1y2+1,z=0,x=2,x=2,y=0,y=151E52EMoments of Inertia In Exercises 53- 56, find Ix,Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) =k (b) =kxyzMoments of Inertia In Exercises 53- 56, find Ix,Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) =(x,y,z)=k (b) =(x,y,z)=k(x2+y2)Moments of Inertia In Exercises 53- 56, find Ix,Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals. (a) =(x,y,z)=k (b) =ky56E57E58E