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All Textbook Solutions for Multivariable Calculus
9RE10RE11REFind a linear approximation to the temperature function T(x, y) in Exercise 11 near the point (6, 4). Then use it to estimate the temperature at the point (5, 3.8).13RE14RE15RE16RE17RE18RE19RE20RE21RE22REIf z = xy + xey/x show that xzx+yzy=xy+z.If z = sin(x + sin t), show that zx2zxt=zt2zx2Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 25. z = 3x2 y2 + 2x, (1, 2, 1)Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 26. z = ex cos y, (0, 0, 1)27RE28REFind equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. 29. sin(xyz) = x + 2y + 3z, (2, 1, 0)Use a computer to graph the surface z = x2 + y4 and its tangent plane and normal line at (1, 1, 2) on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.Find the points on the hyperboloid x2 + 4y2 z2 = 4 where the tangent plane is parallel to the plane 2x + 2y + z = 5.32RE33REThe two legs of a right triangle are measured as 5 m and 12 m with a possible error in measurement of at most 0.2 cm in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and (b) the length of the hypotenuse.35REIf v = x2sin y + yexy, where x = s + 2t and y = st, use the Chain Rule to find v/s and v/t when s = 0 and t = 1.37RE38RE39RE40RE41RE42REFind the gradient of the function f(x,y,z)=x2eyz2.(a) When is the directional derivative of f a maximum? (b) When is it a minimum? (c) When is it 0? (d) When is it half of its maximum value?Find the directional derivative of f at the given point in the indicated direction. 45. f(x, y) = x2ey , (2,0), in the direction toward the point (2. 3)Find the directional derivative of f at the given point in the indicated direction. 46. f(x,y,z)=x2y+x1+z, (1,2, 3), in the direction of v = 2i+,j 2kFind the maximum rate of change of f(x,y)=x2y+y at the point (2, 1). In which direction does it occur?Find the direction in which f(x, y, z) = zexyincreases most rapidly at the point (0, 1, 2). What is the maximum rate of increase?The contour map shows wind speed in knots during Hurricane Andrew on August 24, 1992. Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane.Find parametric equations of the tangent line at the point (2, 2, 4) to the curve of intersection of the surface z = 2x2 y2 and the plane z = 4.Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 51. f(x, y) = x2 xy + y2 + 9x 6y + 1052RE53RE54REFind the absolute maximum and minimum values of f on the set D. 55. f(x, y) = 4xy2 x2y2 xy3; D is the closed triangular region in the .xy-plane with vertices (0, 0), (0, 6), and (6, 0)Find the absolute maximum and minimum values of f on the set D. 55. f(x,y)=ex2y2(x2+2y2); D is the disk x2+y2457RE58REUse Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). 59. f(x, y) = x2y; x2 + y2 = 1Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). 60. f(x,y)=1x+1y; 1x2+1y2=161REUse Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint(s). 62. f(x, y, z) = x2 + 2y2 + 3z2; x + y + z =1, x y + 2z = 263RE64REA pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. II the pentagon has fixed perimeter P, find the lengths of the sides of the pentagon that maximize the area of the pentagon.1PMarine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the Wood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point P(x, y) on the surface of seawater is approximated by C(x, y) = e(x2 + 2y2)/10x where x and y are measured in meters in a rectangular coordinate system with the blood source at the origin. (a) Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source. (b) Suppose a shark is at the point (x0, y0) when it first detects the presence of blood in the water. Find an equation of the shark's path by setting up and solving a differential equation.A long piece of galvanized sheet metal with width w is to be bent into a symmetric form with three straight sides to make a rain gutter. A cross-section is shown in the figure. (a) Determine the dimensions that allow the maximum possible flow; that is, find the dimensions that give the maximum possible cross-sectional area. (b) Would it be better to bend the moral into a gutter with a semicircular cross-section?4PSuppose f is a differentiable function of one variable. Show that all tangent planes to the surface z = xf(y/x) intersect in a common point.6PIf the ellipse x2/a2 + y2/b2 = 1 is to enclose the circle x2 + y2 = 2y, what values of a and b minimize the area of the ellipse?Show that the maximum value of the function f(x,y)=(ax+by+c)2x2+y2+1 is a2 + b2 + c2. Hint: One method for attacking this problem is to use the Cauchy-Schwarz Inequality: |ab||a||b|(a) Estimate the volume of the solid that lies below the surface z = xy and above the rectangle R = {(x, y) | 0 x 6, 0 y 4} Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square. (b) Use the Midpoint Rule to estimate the volume of the solid in part (a).If R = [0, 4] [1, 2], use a Riemann sum with m = 2, n = 3 to estimate the value ofR(1xy2)dA. Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.(a) Use a Riemann sum with m = n = 2 to estimate the value of RxexydA, where R = [0, 2] [0, l]. Take the sample points to be upper right corners. (b) Use the Midpoint Rule to estimate the integral in part (a).(a) Estimate the volume of the solid that lies below the surface z = 1 + x2 + 3y and above the rectangle R = [1, 2] [0, 3]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower left corners. (b) Use the Midpoint Rule to estimate the volume in part (a).Let V be the volume of the solid that lies under the graph of f(x,y)=52x2y2 and above the rectangle given by 2 x 4, 2 y 6. Use the lines x = 3 and y = 4 to divide R into subrectangles. Let L and U be the Riemann sums computed using lower left corners and upper right corners, respectively. Without calculating the numbers V, L, and U, arrange them in increasing order and explain your reasoning.A 20-ft-by-30-ft swimming pool is filled with water. The depth is measured at 5-ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.A contour map is shown for a function f on the square R = [0, 4] [0. 4]. (a) Use the Midpoint Rule with m = n = 2 to estimate the value of Rf(x,y)dA. (b) Estimate the average value of f.The contour map shows the temperature, in degrees Fahrenheit, at 4:00 pm on February 26. 2007, in Colorado. (The state measures 388 mi west to cast and 276 mi south to north.) Use the Midpoint Rule with m = n = 4 to estimate the average temperature in Colorado at that time.Evaluate the double integral by first identifying it as the volume of a solid. 9. R2dA,R={(x,y)|2x6,1y5}Evaluate the double integral by first identifying it as the volume of a solid. 10. R(2x+1)dA,R={(x,y)|0x2,0y4}Evaluate the double integral by first identifying it as the volume of a solid. 11. R(42y)dA,R=[0,1][0,1]The integral R9y2dA, where R = [0, 4] [0, 2], represents the volume of a solid. Sketch the solid.Find 02f(x,y)dxand 03f(x,y)dy 13. f(x, y) = x + 3x2y2Find 02f(x,y)dxand 03f(x,y)dy 14.f(x,y)=yx+2Calculate the iterated integral. 15. 1402(6x2y2x)dydxCalculate the iterated integral. 16. 0101(x+y)2dxdyCalculate the iterated integral. 17. 0112(x+ey)dxdyCalculate the iterated integral. 18. 0/61/2(sinx+siny)dydxCalculate the iterated integral. 19. 330/2(y+y2cosx)dxdyCalculate the iterated integral. 20. 1315lnyxydydxCalculate the iterated integral. 21. 1412(xy+yx)dydxCalculate the iterated integral. 22. 0102yexydxdyCalculate the iterated integral. 23. 030/2t2sin3ddtCalculate the iterated integral. 24. 0101xyx2+y2dydxCalculate the iterated integral. 25. 0101v(u+v2)4dudv26ECalculate the double integral. 27. Rxsec2ydA,R={(x,y)|0x2,0y/4}Calculate the double integral. 28. R(y+xy2)dA,R={(x,y)|0x2,1y2}Calculate the double integral. 29. Rxy2x2+1dA,R={(x,y)|0x1,3y3}Calculate the double integral. 30. Rtan1t2dA,R={(,t)|0/3,0t12}Calculate the double integral. 31. Rxsin(x+y)dA,R=[0,/6][0,/3]32E33E34ESketch the solid whose volume is given by the iterated integral. 35. 0101(4x2y)dxdySketch the solid whose volume is given by the iterated integral. 36. 0101(2x2y2)dydxFind the volume of the solid that lies under the plane 4x + 6y 2z + 15 = 0 and above the rectangle R = {{x, y}| 1 x 2, 1 y 1}.Find the volume of the solid that lies under the hyperbolic paraboloid z = 3y2 x2 + 2 and above the rectangle R = [1, 1] [1, 2].Find the volume of the solid lying under the elliptic paraboloid x2/4 + y2/9 + z = 1 and above the rectangle R = [1, 1] [2, 2].Find the volume of the solid enclosed by the surface z = x2 + xy2 and the planes z = 0, x = 0, x = 5, and y = 2.Find the volume of the solid enclosed by the surface z = 1+ x2yey and the planes z = 0, x = 1, y = 0, and y = 1.Find the volume of the solid in the first octant bounded by the cylinder z = 16 x2 and the plane y = 5.Find the volume of the solid enclosed by the paraboloid z = 2 + x2 + (y 2)2 and the planes z = 1, x = 1, x = 1, y = 0, and y = 4.Graph the solid that lies between the surface z = 2xy/(x2 + 1) and the plane z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4. Then find its volume.Find the average value of f over the given rectangle. 47. f(x, y) = x2y, R has vertices (1, 0), (1, 5), (1, 5), (1, 0)Find the average value of f over the given rectangle. 48. f(x,y)=eyx+ey,R=[0,4][0,1]Use symmetry to evaluate the double integral. 49. Rxy1+x4dA,R={(x,y)|1x1,0y1}50E52EEvaluate the iterated integral. 1. 1s0x(8x2y)dydxEvaluate the iterated integral. 2. 020y2x2ydxdyEvaluate the iterated integral. 3. 010yxey3dxdyEvaluate the iterated integral. 4. 0/20xxsinydydxEvaluate the iterated integral. 5. 010s2cos(s3)dtdsEvaluate the iterated integral. 6. 010ex1+exdwdvEvaluate the double integral. 7. Dyx2+1dA,D={(x,y)|0x4,0yx}Evaluate the double integral. 8. D(2x+y)dA,D={(x,y)|1y2,y1x1}Evaluate the double integral. 9. Dey2dA,D={(x,y)|0y3,0xy}Evaluate the double integral. 10. Dyx2y2dA,D={(x,y)|0x2,0yx}Draw an example of a region that is (a) type I but not type II (b) type II but not type IDraw an example of a region that is (a) both type I and type II (b) neither type I nor type IIExpress D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. 13.DxdA, D is enclosed by the lines y = x, y = 0, x = 1Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. 14. DxydA, D is enclosed by the curves y = x2, y = 3xSet up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why its easier. 15. DydA, D is bounded by y = x 2,.x = y2Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why its easier. 16. Dy2exydA, D is bounded by y = x, y = 4, x = 0Evaluate the double integral. 17.DxcosydA, D is bounded by y = 0. y = x2, x = 1Evaluate the double integral. 18. D(x2+2y)dA, D is bounded by y = x, y = x3, x 0Evaluate the double integral. 19. Dy2dA, D is the triangular region with vertices (0, 1), (1, 2), (4, 1)Evaluate the double integral. 20. DxydA, D is enclosed by the quarter-circle y=1x2,x0, and the axesEvaluate the double integral. 21. D(2xy)dA, D is bounded by the circle with center the origin and radius 2Evaluate the double integral. 22. DydA, D is the triangular region with vertices (0, 0), (1, 1), and (4, 0)Find the volume of the given solid. 23. Under the plane 3x + 2y z = 0 and above the region enclosed by the parabolas y = x2 and x = y2Find the volume of the given solid. 24. Under the surface z = 1+ x2y2 and above the region enclosed by x = y2 and x = 4Find the volume of the given solid. 25. Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2)Find the volume of the given solid. 26. Enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 2Find the volume of the given solid. 27. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4Find the volume of the given solid. 28. Bounded by the planes z = x, y = x, x + y = 2, and z = 0Find the volume of the given solid. 29. Enclosed by the cylinders z = x2, y = x2 and the planes z = 0, y = 4Find the volume of the given solid. 30. Bounded by the cylinder y2 + z2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octantFind the volume of the given solid. 31. Bounded by the cylinder x2 + y2 = 1 and the planes y = z, x = 0, z = 0 in the first octantFind the volume of the given solid. 32. Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r233E34EFind the volume of the solid by subtracting two volumes. 35. The solid enclosed by the parabolic cylinders y = 1 x2, y = x2 1 and the planes x + y + z = 2, 2x + 2y z + 10 = 0Find the volume of the solid by subtracting two volumes. 36. The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + yFind the volume of the solid by subtracting two volumes. 37. The solid under the plane z = 3, above the plane z = y, and between the parabolic cylinders y = x2 and y = 1 x2Find the volume of the solid by subtracting two volumes. 38. The solid in the first octant under the plane z = x + y, above the surface z = xy, and enclosed by the surfaces x = 0, y = 0, and x2 + y2 = 4Sketch the solid whose volume is given by the iterated integral. 0101x(1xy)dydxSketch the solid whose volume is given by the iterated integral. 0101x2(1x)dydxSketch the region of integration and change the order of integration. 010yf(x,y)dxdySketch the region of integration and change the order of integration. 02x24f(x,y)dydxSketch the region of integration and change the order of integration. 0/20cosxf(x,y)dydxSketch the region of integration and change the order of integration. 2204y2f(x,y)dxdySketch the region of integration and change the order of integration. 120lnxf(x,y)dydxSketch the region of integration and change the order of integration. 01arctanx/4f(x,y)dydxEvaluate the integral by reversing the order of integration. 013y3ex2dxdyEvaluate the integral by reversing the order of integration. 01x21ysinydydxEvaluate the integral by reversing the order of integration. 01x1y3+1dydxEvaluate the integral by reversing the order of integration. 02y/21ycos(x31)dxdyEvaluate the integral by reversing the order of integration. 01arcsiny/2cosx1+cos2xdxdyEvaluate the integral by reversing the order of integration. 08y32ex4dxdy57EExpress D as a union of regions of type I or type II and evaluate the integral. 58. DydA59E60E61EFind the averge value of f over the region D. 62. f(x, y) = x sin y, D is enclosed by the curves y = 0, y = x2, and x = 163EIn evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows: Df(x,y)dA=0102yf(x,y)dxdy+1303yf(x,y)dxdy Sketch the region D and express the double integral as an iterated integral with reversed order of integration.Use geometry or symmetry, or both, to evaluate the double integral. 65. D(x+2)dA, D=(x,y)0y9x2Use geometry or symmetry, or both, to evaluate the double integral. 66. DR2x2y2dA, D is the disk with center the origin and radius R67E68E69EA region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R. 1.A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R. 2.A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R. 3.A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write Rf(x,y)dA as an iterated integral, where f is an arbitrary continuous function on R. 4.Sketch the region whose area is given by the integral and evaluate the integral. 5. /43/412rdrdSketch the region whose area is given by the integral and evaluate the integral. 6. /202sinrdrdEvaluate the given integral by changing to polar coordinates. 7. Dx2ydA, where D is the top half of the disk with center the origin and radius 5Evaluate the given integral by changing to polar coordinates. 8. R(2xy)dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = xEvaluate the given integral by changing to polar coordinates. 9. Rsin(x2+y2)dA, where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3Evaluate the given integral by changing to polar coordinates. 10. Ry2x2+y2dA, where R is the region that lies between the circles x2 + y2 = a2 and x2 + y2 = b2 with 0 a bEvaluate the given integral by changing to polar coordinates. 11. Dex2y2dA, where D is the region bounded by the semicircle x=4y2 and the y-axisEvaluate the given integral by changing to polar coordinates. 12. Dcosx2+y2dA, where D is the disk with center the origin and radius 2Evaluate the given integral by changing to polar coordinates. 13. Rarctan(y/x)dA, where R = {(x, y) | 1 x2 + y2 4, 0 y x}Evaluate the given integral by changing to polar coordinates. 14. DxdA, where D is the region in the first quadrant that lies between the circles x2 + y2 = 4 and x2 + y2 = 2xUse a double integral to find the area of the region. 15. One loop of the rose r = cos 3Use a double integral to find the area of the region. 16. The region enclosed by both of the cardioids r = 1 + cos and r = 1 cosUse a double integral to find the area of the region. 17. The region inside the circle (x 1)2 + y2 = 1 and outside the circle x2 + y2 = 1Use a double integral to find the area of the region. 18. The region inside the cardioid r = 1 + cos and outside the circle r = 3 cosUse polar coordinates to find the volume of the given solid. 19. Under the paraboloid z = x2 + y2 and above the disk x2 + y2 25Use polar coordinates to find the volume of the given solid. 20. Below the cone z=x2+y2 and above the ring 1 x2 + y2 4Use polar coordinates to find the volume of the given solid. 21. Below the plane 2x + y + z = 4 and above the disk x2 + y2 1Use polar coordinates to find the volume of the given solid. 22. Inside the sphere x2 + y2 + z2 = 16 and outside the cylinder x2 + y2 = 4Use polar coordinates to find the volume of the given solid. 23. A sphere of radius aUse polar coordinates to find the volume of the given solid. 24. Bounded by the paraboloid z = 1 + 2x2 + 2y2 and the plane z = 7 in the first octantUse polar coordinates to find the volume of the given solid. 25. Above the cone z=x2+y2 and below the sphere x2 + y2 + z2 = 1Use polar coordinates to find the volume of the given solid. 26. Bounded by the paraboloids z = 6 x2 y2 and z = 2x2 + 2y2Use polar coordinates to find the volume of the given solid. 27. Inside both the cylinder x2 + y2 = 4 and the ellipsoid 4x2 + 4y2 + z2 = 64(a) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. (b) Express the volume in part (a) in terms of the height h of the ring. Notice that the volume depends only on h, not on r1 or r2.Evaluate the iterated integral by converting to polar coordinates. 29. 0204x2ex2y2dydxEvaluate the iterated integral by converting to polar coordinates. 30. 0aa2y2a2y2(2x+y)dxdyEvaluate the iterated integral by converting to polar coordinates. 31. 01/23y1y2xy2dxdyEvaluate the iterated integral by converting to polar coordinates. 32. 0202xx2x2+y2dydxExpress the double integral in terms of a single integral with respect to r. Then use your calculator to evaluate the integral correct to four decimal places. 33. De(x2+y2)2dA, where D is the disk with center the origin and radius 1Express the double integral in terms of a single integral with respect to r. Then use your calculator to evaluate the integral correct to four decimal places. 34. Dxy1+x2+y2dA, where D is the portion of the disk x2 + y2 1 that lies in the first quadrantA swimming pool is circular with a 40-ft diameter. The depth is constant along east-west lines and increases linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of water in the pool.An agricultural sprinkler distributes water in a circular pattern of radius 100 ft. It supplies water to a depth of er feet per hour at a distance of r feet from the sprinkler. (a) If 0 R 100, what is the total amount of water supplied per hour to the region inside the circle of radius R centered at the sprinkler? (b) Determine an expression for the average amount of water per hour per square foot supplied to the region inside the circle of radius R.Find the average value of the function f(x,y)=1/x2+y2 on the annular region a2 x2 + y2 b2 where 0 a b.38EUse polar coordinates to combine the sum 1/211x2xxydydx+120xxydydx+2204x2xydydx into one double integral. Then evaluate the double integral.(a) We define the improper integral (over the entire plane 2) I=R2e(x2+y2)dA=e(x2+y2)dydx=limaDae(x2+y2)dA where Da is the disk with radius a and center the origin. Show that e(x2+y2)dA= (b) An equivalent definition of the improper integral in part (a) is R2e(x2+y2)dA=limaSae(x2+y2)dA where Sa is the square with vertices (a, a). Use this to show that ex2dxey2dy= (c) Deduce that ex2dx= (d) By making the change of variable t=2x, show that ex2/2dx=2 (This is a fundamental result for probability and statistics.)41EElectric charge is distributed over the rectangle 0 x 5, 2 y 5 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 + y2 1 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.Find the mass and center of mass of the lamina that occupies the region D and has the given density function . 3. D = {(x, y) | 1 x 3, 1 y 4}; (x, y) = ky2Find the mass and center of mass of the lamina that occupies the region D and has the given density function . 4. D = {(x, y) | 0 x a, 0 y b}; (x, y) = 1 + x2 + y2Find the mass and center of mass of the lamina that occupies the region D and has the given density function . 5. D is the triangular region with vertices (0, 0), (2, 1), (0, 3); (x, y) = x + yFind the mass and center of mass of the lamina that occupies the region D and has the given density function . 6. D is the triangular region enclosed by the lines y = 0, y = 2x, and x + 2y = 1; (x, y) = xFind the mass and center of mass of the lamina that occupies the region D and has the given density function . 7. D is bounded by y = 1 x2 and y = 0; (x, y) = kyFind the mass and center of mass of the lamina that occupies the region D and has the given density function . 8. D is bounded by y = x + 2 and y = x2; (x, y) = kx2Find the mass and center of mass of the lamina that occupies the region D and has the given density function . 9. D is bounded by the curves y = ex, y = 0, x = 0, x = 1; (x, y) = xyFind the mass and center of mass of the lamina that occupies the region D and has the given density function . 10. D is enclosed by the curves y = 0 and y = cos x, /2 x /2; (x, y) = yA lamina occupies the part of the disk x2 + y2 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.12EThe boundary of a lamina consists of the semicircles y=1x2andy=4x2 together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin.Find 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.A lamina occupies the region inside the circle x2 + y2 = 2y but outside the circle x2 + y2 = 1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin.Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 3.Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 6.19E20EA lamina with constant density (x, y) = occupies the given region. Find the moments of inertia lx and Iy and the radii of gyration x and y. 21. The rectangle 0 x b,0 y hA lamina with constant density (x, y) = occupies the given region. Find the moments of inertia lx and Iy and the radii of gyration x and y. 22. The triangle with vertices (0, 0), (b, 0), and (0, h)A lamina with constant density (x, y) = occupies the given region. Find the moments of inertia lx and Iy and the radii of gyration x and y. 23. The part of the disk x2 + y2 a2 in the first quadrantA lamina with constant density (x, y) = occupies the given region. Find the moments of inertia lx and Iy and the radii of gyration x and y. 24. The region under the curve y = sin x from x = 0 to x =27E28E29E(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.32EWhen 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?Find the area of the surface. 1. The part of the plane 5x + 3y - z + 6 = 0 that lies above the rectangle [1, 4] [2, 6]Find the area of the surface. 2. The part of the plane 6x + 4y + 2z = 1 that lies inside the cylinder x2 + y2 = 25Find the area of the surface. 3. The part of the plane 3x + 2y + z = 6 that lies in the first octantFind the area of the surface. 4. The part of the surface 2y + 4z - x2 = 5 that lies above the triangle with vertices (0, 0), (2. 0), and (2, 4)Find the area of the surface. 5. The part of the paraboloid z = 1 x2 y2 that lies above the plane z = 2.Find the area of the surface. 6. The part of the cylinder x2 + z2 = 4 that lies above the square with vertices (0, 0), (1.0), (0, 1). and (1,1)Find the area of the surface. 7. The part of the hyperbolic paraboloid z = y2 x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 48EFind the area of the surface. 9. The part of the surface z = xy that lies within the cylinder x2 + y2 = 1Find the area of the surface. 10. The part of the sphere a x2 + y2 + z2 = 4 that lies above the plane z = 1Find the area of the surface. 11. The part of the sphere x2 + y2- z2 = a2 that lies within the cylinder x2 + y2 = ax and above the xy-planeFind the area of the surface. 12. The part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2+ y2Find 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. 13. The part of the surface z = 1/(1 + x2 + y2) that lies above the disk x2 + y2 114EShow that the area of the part ol the plane z = ax + by + c that projects onto a region D in the xy-plane with area A(D) 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.]The figure shows the surface created when the cylinder y2 + z2 = 1 intersects the cylinder x2 + z2 = 1. Find the area of this surface.Evaluate the integral in Example 1, integrating first with respect to y, then z, and then x. EXAMPLE 1 Evaluate the triple integral Bxyz2dv, where B is the rectangular box given by B=(x,y,z)0x1,1y2,0z3Evaluate the integral E(xy+z2)dv, where E=(x,y,z)0x2,0y1,0z3 using three different orders of integration. 3-8 Evaluate the iterated integral.Evaluate the iterated integral. 3.020z20yz(2xy)dxdydzEvaluate the iterated integral. 4.01y2y0x+y6xydzdxdyEvaluate the iterated integral. 5. 1202z0lnxxeydydxdzEvaluate the iterated integral. 6. 010101z2zy+1dxdzdyEvaluate the iterated integral. 7.00101z2zsinxdydzdxEvaluate the iterated integral. 8. 010102x2y2xyezdzdydxEvaluate the triple integral. 9. EydV, where E=(x,y,z)0x3,0yx,xyzx+yEvaluate the triple integral. 10.EezydV, where E=(x,y,z)0y1,yx1,0zxyEvaluate the triple integral. 11. Ezx2+z2dV, where E=(x,y,z)1y4,yz4,0xzEvaluate the triple integral. 12. EsinydV, where E lies below the plane z = x and above the triangular region with vertices (0, 0, 0). (, 0, 0). and (0, , 0)Evaluate the triple integral. 13. E6xydV, where E lies under the plane z = 1 + x + y and above the region in the .xy-plane bounded by the curves y = x,y = 0. and x = 1Evaluate the triple integral. 14. E(xy)dV, where E lies enclosed by the surface z = x2 1, z = 1 x2, y = 0 and y = 2Evaluate the triple integral. 15. Ty2dV. where T is the solid tetrahedron with vertices (0, 0,0), (2, 0, 0). (0, 2, 0). and (0, 0, 2)Evaluate the triple integral. 16. TxzdV, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1)Evaluate the triple integral. 17. ExdV, where E is bounded by the paraboloid x 4y2 + 4z2 and the plane x = 4Evaluate the triple integral. 18. EzdV, where E is bounded by the cylinder y2 + z2 = 9 and the planes x = 0, y = 3x, and z = 0 in the first octantUse a triple integral to find the volume of the given solid. 19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4Use a triple integral to find the volume of the given solid. 20. The solid enclosed by the paraboloids y = x2 + z2 and y = 8 - x2 - z2Use a triple integral to find the volume of the given solid. 21. The solid enclosed by the cylinder y = x2 and the planes z = 0 and y + z = 1Use a triple integral to find the volume of the given solid. 22. The solid enclosed by the cylinder x2 + z2 = 4 and the planes y = -1 and y + z = 4Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub-boxes of equal size. 25. Bcos(xyz)dV,where B=(x,y,z)0x1,0y1,0z1Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub-boxes of equal size. 26. Bxexyzdv,where, where B=(x,y,z)0x4,0y1,0z2Sketch the solid whose volume is given by the iterated integral. 27. 0101x022xdydzdxSketch the solid whose volume is given by the iterated integral. 28.0202y04y2dxdzdyExpress the integralEf(x,y,z)dV, as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. 29. y = 4 - x2 - 4z2, y = 0Express 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. 30. y2 + z2 = 9, x = -2, x = 2Express 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. 31. y =x2, z = 0, y + 2z = 4Express 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. 32. x = 2, y = 2, z = 0, x + y 2z = 2The 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.