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All Textbook Solutions for Calculus (MindTap Course List)
38EUse polar coordinates to combine the sum 1/211x2xxydydx+120xxydydx+ 2204x2xydydx into the double integral. Then evaluate the double integral.40EUse the result of Exercise 40 partc to evaluate the following integrals. a 0x2ex2dxb 0xexdxElectric charge is distributed over the rectangle 0x5, 2y5 so that the charge density at (x,y) is (x,y)=2x+4y measured in coulombs per square meter. Find the total charge on the rectangle.Electric charge is distributed over the disk x2+y21 so that the charge density at (x,y) is (x,y)=x2+y2 measured in coulombs per square meter. Find the total charge on the disk.3E4E5E310 Find the mass and center of mass of the lamina that occupies the region D and has the given density function . D is the triangular region enclosed by the lines y=0,y=2x, and x+2y=1;(x,y)=x7E8E9E10EA lamina occupies the part of the disk x2+y21 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.12E13EFind the center of mass of the lamina in Exercise 13 if the density at any point is inversely proportional to its distance from the origin.Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length a if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.16E17E18EFind the moments of inertia Ix,Iy,I0 for the lamina of Exercise 15.20E21E22E23E24E25E26E27E28ESuppose X and Y are random variables with joint density function f(x,y)={0.1e(0.5x+0.2y)ifx0,y00otherwise a Verify that f is indeed a joint density function. b Find the following probabilities. i P(Y1) ii P(X2,Y4) c Find the expected values of X and Y.a A lamp has two bulbs, each of a type with average lifetime 1000 hours. Assuming that we can model the probability of failure of a bulb by an exponential density function with mean =1000, find the probability that both of the lamps bulbs fail within 1000 hours. b Another lamp has just one bulb of the same type as in part a. If one bulb bums out and is replaced by a bulb of the same type, find the probability that the two bulbs fail within a total of 1000 hours.Suppose that X and Y are independent random variables, where X is normally distributed with mean 45 and standard deviation 0.5 and Y is normally distributed with mean 20 and standard deviation 0.1. a Find P(40X50,20Y25). b Find P(4(X45)2+100(Y20)22).Xavier and Yolanda both have classes that end at noon and they agree to meet every day after class. They arrive at the coffee shop independently. Xaviers arrival time is X and Yolandas arrival time is Y, where X and Y are measured in minutes after noon. The individual density functions are f1(x)={exifx00ifx0f2(y)={150yif0y100otherwise Xavier arrives sometime after noon and is more likely to arrive promptly than late. Yolanda always arrives by 12:10 pm and is more likely to arrive late than promptly. After Yolanda arrives, shell wait for up to half an hour for Xavier, but he wont wait for her. Find the probability that they meet.When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them. Consider a circular city of radius 10 miles in which the population is uniformly distributed. For an uninfected individual at a fixed point A(x0,y0), assume that the probability function is given by f(P)=120[20d(P,A)] where d(P,A) denotes the distance between points P and A. a Suppose the exposure of a person to the disease is the sum of the probabilities of catching the disease from all members of the population. Assume that the infected people are uniformly distributed throughout the city, with k infected individuals per square mile. Find a double integral that represents the exposure of a person residing at A. b Evaluate the integral for the case in which A is the center of the city and for the case in which A is located on the edge of the city. Where would you prefer to live?112 Find the area of the surface. The part of the plane 5x+3yz+6=0 that lies above the rectangle [1,4][2,6]112 Find the area of the surface. The part of the plane 6x+4y+2z=1 that lies inside the cylinder x2+y2=253E4E5E112 Find the area of the surface. The part of the cylinder x2+y2=4 that lies above the square with vertices (0,0),(1,0),(0,1), and (1,1)7E112 Find the area of the surface. The surface z=23(x3/2+y3/2),0x1,0y1112 Find the area of the surface. The part of the surface z=xy that lies within the cylinder x2+y2=1112 Find the area of the surface. The part of the sphere x2+y2+z2=4 that lies above the plane z=1112 Find the area of the surface. The part of the sphere x2+y2+z2=a2 that lies within the cylinder x2+y2=ax and above the xy-plane112 Find the area of the surface. The part of the sphere x2+y2+z2=4z that lies inside the paraboloid z=x2+y21314 Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z=1/(1+x2+y2) that lies above the disk x2+y211314 Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z=cos(x2+y2) that lies inside the cylinder x2+y2=1.a Use the Midpoint Rule for double integrals see Section 15.1 with four squares to estimate the surface area of the portion of the paraboloid z=x2+y2 that lies above the square [0,1][0,1]. b Use a computer algebra system to approximate the surface area in part a to four decimal places. Compare with the answer to part a.16EFind the exact area of the surface z=1+2x+3y+4y2,1x4,0y1.18EFind, to four decimal places, the area of the part of the surface z=1+x2y2 that lies above the disk x2+y21.20EShow that the area of the part of the plane z=ax+by+c that projects onto a region D in the xy-plane with area AD is a2+b2+1A(D).22EFind the area of the finite part of the paraboloid y=x2+z2 cut off by the plane y=25. Hint: Project the surface onto the xz-plane.24EEvaluate the integral in Example 1, integrating first with respect to y, then z, and then x.2E38 Evaluate the iterated integral. 020z20yz(2xy)dxdydz38 Evaluate the iterated integral. 01y2y0x+y6xydzdxdy5E38 Evaluate the iterated integral. 010101z2zy+1dxdzdy38 Evaluate the iterated integral. 00101z2zsinxdydzdx38 Evaluate the iterated integral. 010102x2y2xyezdzdydx9E918 Evaluate the triple integral. Eez/ydV, where E={(x,y,z)0y1,yx1,0zxy}11E918 Evaluate the triple integral. EsinydV, where E lies below the plane z=x and above the triangular region with vertices (0,0,0),(,0,0), and (0,,0)13E918 Evaluate the triple integral. E(xy)dV, whereE is enclosed by the surface z=x21,z=1x2,y=0, and y=215E918 Evaluate the triple integral. TxzdV, where T is the solid tetrahedron with vertices 0, 0, 0, 1, 0, 1, 0, 1, 1, and 0, 0, 117E918 Evaluate the triple integral. EzdV, where E is bounded by the cylinder y2+z2=9 and the planes x=0,y=3x, and z=0 in the first octant1922 Use a triple integral to find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane 2x+y+z=41922 Use a triple integral to find the volume of the given solid. The solid enclosed by the paraboloids y=x2+z2 and y=8x2z221E1922 Use a triple integral to find the volume of the given solid. The solid enclosed by the cylinder x2+z2=4 and the planes y=1 and y+z=4a Express the volume of the wedge in the first octant that is cut from the cylinder y2+z2=1 by the planes y=x and x=1 as a triple integral. b Use either the Table of Integrals on Reference Pages 610 or a computer algebra system to find the exact value of the triple integral in part a.a In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where f(x,y,z)is evaluated at the center (xi,yj,zk) of the box Bijk. Use the Midpoint Rule to estimate Bx2+y2+z2dV, where B is the cube defined by 0x4,0y4,0z4. Divide B into eight cubes of equal size. b Use a computer algebra system to approximate the integral in part a correct to the nearest integer. Compare with the answer to part a.2526 Use the Midpoint Rule for triple integrals Exercise 24 to estimate the value of the integral. Divide B into eight sub-boxes of equal size. Bcos(xyz)dV, where B={(x,y,z)0x1,0y1,0z1}26E2728 Sketch the solid whose volume is given by the iterated integral. 0101x022zdydzdx2728 Sketch the solid whose volume is given by the iterated integral. 0202y04y2dxdzdy2932 Express the integral Ef(x,y,z)dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. y=4x24z2,y=030E31E32EThe figure shows the region of integration for the integral 01x101yf(x,y,z)dzdydx Rewrite this integral as an equivalent iterated integral in the five other orders.The figure shows the region of integration for the integral 0101x201xf(x,y,z)dydzdx Rewrite this integral as an equivalent iterated integral in the five other orders.3536 Write five other iterated integrals that are equal to the given iterated integral. 01y10yf(x,y,z)dzdxdy3536 Write five other iterated integrals that are equal to the given iterated integral. 01y10zf(x,y,z)dxdzdy37E38E3942 Find the mass and center of mass of the solid E with the given density function . E lies above the xy-plane and below the paraboloid z=1x2y2;(x,y,z)=33942 Find the mass and center of mass of the solid E with the given density function . E is bounded by the parabolic cylinder z=1y2 and the planes x+z=1,x=0,andz=0;(x,y,z)=441E42E43E4346 Assume that the solid has constant density k. Find the moments of inertia for a rectangular brick with dimensions a, b, and c and mass M if the center of the brick is situated at the origin and the edges are parallel to the coordinate axes.45E4346 Assume that the solid has constant density k. Find the moment of inertia about the z-axis of the solid cone x2+y2zh.47E48ELet E be the solid in the first octant bounded by the cylinder x2+y2=1 and the planes y=z,x=0,andz=0 with the density function (x,y,z)=1+x+y+z. Use a computer algebra system to find the exact values of the following quantities for E. a The mass b The center of mass c The moment of inertia about the z-axisIf E is the solid of Exercise 18 with density function (x,y,z)=x2+y2, find the following quantities, correct to three decimal places. a The mass b The center of mass c The moment of inertia about the z-axisThe joint density function for random variables X, Y, and Z is f(x,y,z)=cxyz if 0x2,0y2,0z2, and f(x,y,z)=0 otherwise. a Find the value of the constant C. b Find P(X1,Y1,Z1). c Find. P(X+Y+Z1).52E5354 The average value of a function f(x,y,z)over a solid region E is defined to be fave=1v(E)Ef(x,y,z)dV where V(E) is the volume of E. For instance, if is a density function, then ave is the average density of E. Find the average value of the function f(x,y,z)=xyzover the cube with side length L that lies in the first octant with one vertex at the origin and edges parallel to the coordinate axes.5354 The average value of a function f(x,y,z)over a solid region E is defined to be fave=1v(E)Ef(x,y,z)dV where V(E) is the volume of E. For instance, if is a density function, then ave is the average density of E. Find the average height of the points in the solid hemisphere x2+y2+z21,z0.a Find the region E for which the triple integral E(1x22y23z2)dV is a maximum. b Use a computer algebra system to calculate the exact maximum value of the triple integral in part a.12 Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. a (4,/3,2) b (2,/2,1)2E3E4E56 Describe in words the surface whose equation is given. r=26E78 Identify the surface whose equation is given. r2+z2=48E9E910 Write the equations in cylindrical coordinates. a 2x2+2y2z2=4 b 2xy+z=111E1112 Sketch the solid described by the given inequalities. 0/2,rz213E14E15E1516 Sketch the solid whose volume is given by the integral and evaluate the integral. 02020rrdzddr1728 Use cylindrical coordinates. Evaluate Ex2+y2dV where E is the region that lies inside the cylinder x2+y2=16 and between the planes z=5 and z=4.18E1728 Use cylindrical coordinates. Evaluate E(x+y+z)dV where E is the solid in the first octant that lies under the paraboloid z=4x2y2.20E1728 Use cylindrical coordinates. Evaluate Ex2dV where E is the solid that lies within the cylinder x2+y2=1, above the plane z=0, and below the cone z2=4x2+4y2.22E23E24E1728 Use cylindrical coordinates. a Find the volume of the region E that lies between the paraboloid z=24x2y2 and the cone z=2x2+y2. b Find the centroid of E the center of mass in the case where the density is constant.1728 Use cylindrical coordinates. a Find the volume of the solid that the cylinder r=acos cuts out of the sphere of radius a centered at the origin. b Illustrate the solid of part a by graphing the sphere and the cylinder on the same screen.1728 Use cylindrical coordinates. Find the mass and center of mass of the solid S bounded by the paraboloid z=4x2+4y2 and the plane z=a(a0) if S has constant density K.28E2930 Evaluate the integral by changing to cylindrical coordinates. 224y24y2x2+y22xzdzdxdy2930 Evaluate the integral by changing to cylindrical coordinates. 3309x209x2y2x2+y2dzdydxWhen studying the formation of mountain ranges, geologists estimate the amount of work required to lift a mountain from sea level. Consider a mountain that is essentially in the shape of a right circular cone. Suppose that the weight density of the material in the vicinity of a point P is g(P) and the height is h(P). a Find a definite integral that represents the total work done in forming the mountain. b Assume that Mount Fuji in Japan is in the shape of a right circular cone with radius 62, 000 ft, height 12, 400 ft, and density a constant 200 lb/ft3. How much work was done in forming Mount Fuji if the land was initially at sea level?12 Plot the point whose spherical coordinates are given. Then find the rectangular coordinates of the point. a (6,/3,/6) b (3,/2,3/4)2E3E4E56 Describe in words the surface whose equation is given. =/356 Describe in words the surface whose equation is given. p23p+2=07E8E9E910 Write the equation in spherical coordinates. a z=x2+y2 b z=x2y21114 Sketch the solid described by the given inequalities. 1, 0/6, 012E13E14EA solid lies above the cone z=x2+y2 and below the sphere x2+y2+z2=z. Write a description of the solid in terms of inequalities involving spherical coordinates.a Find inequalities that describe a hollow ball with diameter 30 cm and thickness 0.5 cm. Explain how you have positioned the coordinate system that you have chosen. b Suppose the ball is cut in half. Write inequalities that describe one of the halves.1718 Sketch the solid whose volume is given by the integral and evaluate the integral. 0/60/2032sinddd18E1920 Set up the triple integral of an arbitrary continuous function f(x,y,z) in cylindrical or spherical coordinates over the solid shown.1920 Set up the triple integral of an arbitrary continuous function f(x,y,z) in cylindrical or spherical coordinates over the solid shown.2134 Use spherical coordinates. Evaluate B(x2+y2+z2)2dV, where B is the ball with center the origin and radius 5.22E23E24E2134 Use spherical coordinates. Evaluate Exex2+y2+z2dV, where E is the portion of the unit ball x2+y2+z21 that lies in the first octant.26E2134 Use spherical coordinates. Find the volume of the part of the ball a that lies between the cones =/6 and =/3.2134 Use spherical coordinates. Find the average distance from a point in a ball of radius a to its center.2134 Use spherical coordinates. a Find the volume of the solid that lies above the cone =/3 and below the sphere =4cos. b Find the centroid of the solid in part a.2134 Use spherical coordinates. Find the volume of the solid that lies within the sphere x2+y2+z2=4, above the xy-plane, and below the cone z=x2+y2.31E2134 Use spherical coordinates. Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base. a Find the mass of H. b Find the center of mass of H. c Find the moment of inertia of H about its axis.2134 Use spherical coordinates. a Find the centroid of a solid homogeneous hemisphere of radius a. b Find the moment of inertia of the solid in part a about a diameter of its base.34E3540 Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume and centroid of the solid E that lies above the cone z=x2+y2 and below the sphere x2+y2+z2=1.36E3540 Use cylindrical or spherical coordinates, whichever seems more appropriate. A solid cylinder with constant density has base radius a and height h. a Find the moment of inertia of the cylinder about its axis. b Find the moment of inertia of the cylinder about a diameter of its base.3540 Use cylindrical or spherical coordinates, whichever seems more appropriate. A solid right circular cone with constant density has base radius a and height h. a Find the moment of inertia of the cone about its axis. b Find the moment of inertia of the cone about a diameter of its base.3540 Use cylindrical or spherical coordinates, whichever seems more appropriate. Evaluate EzdV, where E lies above the paraboloid z=x2+y2 and below the plane z=2y. Use either the Table of Integrals on Reference Pages 610 or a computer algebra system to evaluate the integral.3540 Use cylindrical or spherical coordinates, whichever seems more appropriate. a Find the volume enclosed by the torus =sin. b Use a computer to draw the torus.4143 Evaluate the integral by changing to spherical coordinates. 0101x2x2+y22x2y2xydzdydx4143 Evaluate the integral by changing to spherical coordinates. aaa2y2a2y2a2x2y2a2x2y2(x2z+y2z+z3)dzdxdy4143 Evaluate the integral by changing to spherical coordinates. 224x24x224x2y22+4x2y2(x2+y2+z2)3/2dzdydxA model for the density of the earths atmosphere near its surface is =619.090.000097 where the distance from the center of the earth is measured in meters and is measured in kilograms per cubic meter. If we take the surface of the earth to be a sphere with radius 6370 km, then this model is a reasonable one for 6.3701066.375106. Use this model to estimate the mass of the atmosphere between the ground and an altitude of 5 km.Use a graphing device to draw a silo consisting of a cylinder with radius 3 and height 10 surmounted by a hemisphere.The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates ,, as follows. We take the origin to be the center of the earth and the positive z-axis to pass through the North Pole. The positive x-axis passes through the point where the prime meridian the meridian through Greenwich, England intersects the equator. Then the latitude of P is =90 and the longitude is =360. Find the great-circle distance from Los Angeles lat. 34.06 N, long. 118.25 W to Montral lat. 45.50 N, long. 73.60 W. Take the radius of the earth to be 3960 mi. A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.The surfaces =1+15sinmsinn have been used as models for tumors. The bumpy sphere with m=6 and n=5 is shown. Use a computer algebra system to find the volume it encloses.48Ea Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r2+z2=a2 and below by the cone z=rcot0(or=0), where 00/2, is V=2a33(1cos0) b Deduce that the volume of the spherical wedge given by 12,12,12, is V=23133(cos1cos2)(21) c Use the Mean Value Theorem to show that the volume in part b can be written as V=~2sin~ Where ~ lies between 1 and 2, ~ lies between 1 and 2=21,=21,and=2116 Find the Jacobian of the transformation. x=2u+v,y=4uv2E3E4E5E16 Find the Jacobian of the transformation. x=u+vw,y=v+wu,z=w+uv7E710 Find the image of the set S under the given transformation. S is the square bounded by the lines u=0,u=1,v=0,v=1;x=v,y=u(1+v2)710 Find the image of the set S under the given transformation. S is the triangular region with vertices (0,0),(1,1),(0,1); x=u2,y=v10E11E1114 A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R is the parallelogram with vertices (0,0),(4,3),(2,4),(2,1)13E1114 A region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v-axes. R is bounded by the hyperbolas y=1/x,y=4/x and the lines y=x,y=4x in the first quadrant15E1520 Use the given transformation to evaluate the integral. R(4x+8y)dA, where R is the parallelogram with vertices (1,3),(1,3), (3,1) and (1,5); x=14(u+v), y=14(v3u)1520 Use the given transformation to evaluate the integral. Rx2dA, where R is the region bounded by the ellipse 9x2+4y2=36; x=2u,y=3v18E1520 Use the given transformation to evaluate the integral. RxydA, where R is the region in the first quadrant bounded by the lines y=x and y=3x and the hyperbolas xy=1,xy=3;x=u/v,y=v1520 Use the given transformation to evaluate the integral. Ry2dA, where R is the region bounded by the curves xy=1,xy=2,xy2=1,xy2=2; u=xy,v=xy2. Illustrate by using a graphing calculator or computer to draw R.a Evaluate EdV, where E is the solid enclosed by the ellipsoid x2/a2+y2/b2+z2/c2=1. Use the transformation x=au,y=bv,z=cw. b The earth is not a perfect sphere; rotation has resulted in flattening at the poles. So the shape can be approximated by an ellipsoid with a=b=6378km and c=6356km. Use part a to estimate the volume of the earth. c If the solid of part a has constant density k, find its moment of inertia about the z-axis.An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy=a,xy=b and two adiabatic curves xy1.4=c,xy1.4=d, where 0ab and 0cd Compute the work done by determining the area of R.23E2327 Evaluate the integral by making an appropriate change of variables. R(x+y)ex2y2dA, where R is the rectangle enclosed by the lines xy=0,xy=2,x+y=0, and x+y=32327 Evaluate the integral by making an appropriate change of variables. Rcos(yxy+x)dA, where R is the trapezoidal region with vertices (1,0),(2,0),(0,2), and (0,1)2327 Evaluate the integral by making an appropriate change of variables. Rsin(9x2+4y2)dA, where R is the region in the first quadrant bounded by the ellipse 9x2+4y2=12327 Evaluate the integral by making an appropriate change of variables. Rex+ydA, where R is given by the inequality |x|+|y|1Let f be continuous on [0,1] and let R be the triangular region with vertices (0,0),(1,0), and (0,1). Show that Rf(x+y)dA=01uf(u)du1CC2CC3CC4CC5CC6CCa Write the definition of the triple integral of f over a rectangular box B. b How do you evaluate Bf(x,y,z)dV? c How do you define Ef(x,y,z)dV if E is a bounded solid region that is not a box? d What is type 1 solid region? How do you evaluate Ef(x,y,z)dV if E is such a region? e What is a type 2 solid region? How do you evaluate Ef(x,y,z)dV if E is such a region? f What is a type 3 solid region? How do you evaluate Ef(x,y,z)dV if E is such a region?8CC9CC10CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQA contour map is shown for a function f on the square R=[0,3][0,3]. Use a Riemann sum with nine terms to estimate the value of Rf(x,y)dA. Take the sample points to be the upper right corners of the squares.2E3E4E5E6E7E8E9E910 Write Rf(x,y)dA as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R.11E12E13E14E15ESketch the solid consisting of all points with spherical coordinates (,,) such that 0/2,0/6, and 02cos.Describe the region whose area is given by the integral 0/20sin2rdrdDescribe the solid whose volume is given by the integral 0/20/2122sinddd and evaluate the integral.1920 Calculate the iterated integral by first reversing the order of integration. 01x1cos(y2)dydx1920 Calculate the iterated integral by first reversing the order of integration. 01y1yex2x3dxdy21E22E23E24E2134 Calculate the value of the multiple integral. DydA, where D is the region in the first quadrant bounded by the parabolas x=y2 and x=8y22134 Calculate the value of the multiple integral. DydA, where D is the region in the first quadrant that lies above the hyperbola xy=1 and the line y=x and below the line y=22134 Calculate the value of the multiple integral. D(x2+y2)3/2dA where D is the region in the first quadrant bounded by the lines y=0 and y=3x and the circle x2+y2=928E29E2134 Calculate the value of the multiple integral. TxydV, where T is the solid tetrahedron with vertices (0,0,0),(13,0,0),(0,1,0), and (0,0,1)2134 Calculate the value of the multiple integral. Ey2z2dV, where E is bounded by the paraboloid x=1y2z2 and the plane x=032E2134 Calculate the value of the multiple integral. EyzdV, where E lies above the plane z=0, below the plane z=y, and inside the cylinder x2+y2=42134 Calculate the value of the multiple integral. Hz3x2+y2+z2dV, where H is the solid hemisphere that lies above the xy-plane and has center the origin and radius 135E3540 Find the volume of the given solid. Under the surface z=x2y and above the triangle in the xy-plane with vertices (1,0),(2,1), and (4,0)37E38E39E40EConsider a lamina that occupies the region D bounded by the parabola x=1y2 and the coordinate axes in the first quadrant with density function (x,y)=y. a Find the mass of the lamina. b Find the center of mass. c Find the moments of inertia and radii of gyration about the x- and y-axes.42Ea Find the centroid of a solid right circular cone with height h and base radius a. Place the cone so that its bas is in the xy-plane with center the origin and its axis along the positive z-axis. b If the cone has density function (x,y,z)=x2+y2, find the moment of inertia of the cone about its axis the z-axis.Find the area of the part of the cone z2=a2(x2+y2) between the planes z=1 and z=2.45EGraph the surface z=xsiny,3x3,y, and find its surface area correct to four decimal places.47E48E49E50E51E52E53E54E55EUse the transformation x=u2,y=v2,z=w2 to find the volume of the region bounded by the surface x+y+z=1 and the coordinate planes.57E