Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Calculus: Early Transcendentals

(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.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?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 = 25 Find the area of the surface. 3. The part of the plane 3x + 2y + z = 6 that lies in the first octant Find 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 = 4 Find the area of the surface. 8. The surface z=23(x32+y32),0x1,0y1 Find the area of the surface. 9. The part of the surface z = xy that lies within the cylinder x2 + y2 = 1 Find the area of the surface. 10. The part of the sphere a x2 + y2 + z2 = 4 that lies above the plane z = 1 Find 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-plane Find the area of the surface. 12. The part of the sphere x2 + y2 + z2 = 4z that lies inside the paraboloid z = x2+ y2 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. 13. The part of the surface z = 1/(1 + x2 + y2) that lies above the disk x2 + y2 1 14E 21E If you attempt to use Formula 2 to find the area of the top half of the sphere x2 + y2 + z2 = a2, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circle x2 + y2 = a2. However, the integral can be computed as the limit of the integral over the disk x2 + y2 t2 as t a-. Use this method to show that the area of a sphere of radius a is 4a2.Find 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)0x1,1y2,0z3 Evaluate 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)dxdydz Evaluate the iterated integral. 4.01y2y0x+y6xydzdxdy Evaluate the iterated integral. 5. 1202z0lnxxeydydxdz Evaluate the iterated integral. 6. 010101z2zy+1dxdzdy Evaluate the iterated integral. 7.00101z2zsinxdydzdx Evaluate the iterated integral. 8. 010102x2y2xyezdzdydx Evaluate the triple integral. 9. EydV, where E=(x,y,z)0x3,0yx,xyzx+y Evaluate the triple integral. 10.EezydV, where E=(x,y,z)0y1,yx1,0zxy Evaluate the triple integral. 11. Ezx2+z2dV, where E=(x,y,z)1y4,yz4,0xz Evaluate 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 = 1 Evaluate the triple integral. 14. E(xy)dV, where E lies enclosed by the surface z = x2 1, z = 1 x2, y = 0 and y = 2 Evaluate 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 = 4 Evaluate 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 octant Use 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 = 4 Use 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 - z2 Use 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 = 1 Use 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 = 4 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. 25. Bcos(xyz)dV,where B=(x,y,z)0x1,0y1,0z1 26E Sketch the solid whose volume is given by the iterated integral. 27. 0101x022xdydzdx Sketch the solid whose volume is given by the iterated integral. 28.0202y04y2dxdzdy Express 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 = 0 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. 30. y2 + z2 = 9, x = -2, x = 2 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. 31. y =x2, z = 0, y + 2z = 4 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. 32. x = 2, y = 2, z = 0, x + y 2z = 2 The 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.Write five other iterated integrals that are equal to the given iterated integral. 35. 01y10yf(x,y,z)dzdxdy Write five other iterated integrals that are equal to the given iterated integral. 36. 01y10zf(x,y,z)dxdzdy Evaluate the triple integral using only geometric interpretation and symmetry. 37.c(4+5x2yz2)dV, where C is the cylindrical region x2+y24,2z2 Evaluate the triple integral using only geometric interpretation and symmetry. 38. B(z3+siny+3)dV, where B is the unit ball x2 + y2 + z2 1 Find the mass and center of mass of the solid E with the given density function . 39. E lies above the xy-plane and below the paraboloid z = 1 x2 y2; (x, y, z) = 3 Find the mass and center of mass of the solid R with the given density function . 40. E is bounded by the parabolic cylinder z = 1 y2 and the planes x + z = 1, x = 0, and z = 0; (x, y, z) = 4 Find the mass and center of mass of the solid E with the given density function . 41. E. is the cube given by 0 x a, 0 y a, 0 z a; (x, y, z) = x2 + y2 + z2 Find the mass and center of mass of the solid F. with the given density function . 42. E is the tetrahedron bounded by the planes x = 0, y = 0, z = 0, x + y + z = 1; (x, y, z) = y Assume that the solid has constant density k. 43. Find the moments of inertia for a cube with side length L if one vertex is located at the origin and three edges lie along the coordinate axes.Assume that the solid has constant density k. 44. 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 arc parallel to the coordinate axes.45E Assume that the solid has constant density k. 46. Find the moment of inertia about the z-axis of the solid cone x2+y2zh.47E Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis. 48. The hemisphere x2 + y2 + z2 1, z 0; (x, y, z) = x2+y2+z2 51E 52E 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. 53. Find the average value of the function f(x, y, z) = xyz over 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.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. 54. Find the average height of the points in the solid hemisphere x2 + y2 + z2 1, z 0.Plot the point whose cylindrical coordinates are given. Then find the rectangular coordinates of the point. 1. (a) (4, /3, 2) (b) (2, /2. 1)2E Change from rectangular to cylindrical coordinates. 3. (a) (1, 1, 1) (b) (2,23,3)Change from rectangular to cylindrical coordinates. 4. (a) (2,2,1) (b) (2, 2, 2)Describe in words the surface whose equation is given. 5. r = 2 Describe in words the surface whose equation is given. 6. = /6 Identify the surface whose equation is given. 7. r2 + z2 = 4 Identify the surface whose equation is given. 8. r = 2 sin Write the equations in cylindrical coordinates. 9. (a) x2 x + y2 + z2 = 1 (b) z = x2 y2 Write the equations in cylindrical coordinates. 10. (a) 2x2 + 2y2 z2 = 4 (b) 2x y + z = 1 Sketch the solid described by the given inequalities. 11. r2 z 8 r2 Sketch the solid described by the given inequalities. 12. 0 /2, r z 2 A cylindrical shell is 20 cm long, with inner radius 6 cm and outer radius 7 cm. Write inequalities that describe the shell in an appropriate coordinate system. Explain how you have positioned the coordinate system with respect to the shell.Use a graphing device to draw the solid enclosed by the paraboloids z = x2 + y2 and z = 5 x2 y2.Sketch the solid whose volume is given by the integral and evaluate the integral. 15./2/2020r2rdzdrd Sketch the solid whose volume is given by the integral and evaluate the integral. 16. 02020rrdzddr Use cylindrical coordinates. 17. 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.Use cylindrical coordinates. 18. EvaluateEZdV, where E is enclosed by the paraboloid z = x2 + y2 and the plane z = 4.Use cylindrical coordinates. 19. Evaluate E(x+y+z)dV, where E is the solid in the first octant that lies under the paraboloid z = 4 x2 y2.Use cylindrical coordinates. 20. EvaluateE(xy)dV, where E is the solid that lies between the cylinders x2 + y2 = 1 and .x2 + y2 = 16, above the xy-plane, and below the plane z = y + 4.Use cylindrical coordinates. 21. Evaluate Ex2dV, where E is the solid that lies within the cylinder .x2 + y2 = l, above the plane z = 0, and below the cone z2 = 4x2 + 4y2.Use cylindrical coordinates. 22. Find the volume of the solid that lies within both the cylinder x2 + y2 = 1 and the sphere x2 + y2 + z2 = 4.Use cylindrical coordinates. 23. Find the volume of the solid that is enclosed by the cone z = x2+y2and the sphere x2 + y2 + z2 = 2.Use cylindrical coordinates. 24. Find the volume of the solid that lies between the paraboloid z = x2 + y2 and the sphere x2 + y2 + z2 = 2.Use cylindrical coordinates. 25. (a) Find the volume of the region E that lies between the paraboloid z = 24 - x2 - y2 and the cone Z = 2x2+y2. (b) Find the centroid of E (the center of mass in the case where the density is constant).Use cylindrical coordinates. 26. (a) Find the volume of the solid that the cylinder r = a cos 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.Use cylindrical coordinates. 27. Find the mass and center of mass of the solid S hounded by the paraboloid z = 4x2 + 4y2 and the plane z = a (a 0) if S has constant density K.Use cylindrical coordinates. 28. Find the mass of a ball B given by x2 + y2 + z2 a2 if the density at any point is proportional to its distance from the z-axis.Evaluate the integral by changing to cylindrical coordinates. 29. 224y24y2x2+y22xzdzdxdy Evaluate the integral by changing to cylindrical coordinates. 30.3309x209x2y2x2+y2dz dy dx When 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?1E 2E 3E Change from rectangular to spherical coordinates. 4. (a) (1, 0, 3) (b) (3, 1, 23)Describe in words the surface whose equation is given. 5. = /3 Describe in words the surface whose equation is given. 6. 2 3 + 1 = 0 Identify the surface whose equation is given. 7. cos = 1 Identify the surface whose equation is given. 8. = cos Write the equation in spherical coordinates. 9. (a) x2 + y2 + z2 = 9 (b) x2 y2 z2 = 1 Write the equation in spherical coordinates. 10. (a) z = x2 + y2 (b) z = x2 y2 Sketch the solid described by the given inequalities. 11. 1, 0 /6, 0 Sketch the solid described by the given inequalities. 12. 1 2, /2 Sketch the solid described by the given inequalities. 13. 2 4, 0 /3, 0 Sketch the solid described by the given inequalities. 14. 2, csc A 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 describes one of the halves.Sketch the solid whose volume is given by the integral and evaluate the integral. 17. 0/60/2232 sin d d d Sketch the solid whose volume is given by the integral and evaluate the integral. 18. 0/4022sec2 sin d d d Set up the triple integral of an arbitrary of an arbitrary continuous function f(x, y, z) in cylindrical or spherical coordinates over the solid shown.Set up the triple integral of an arbitrary of an arbitrary continuous function f(x, y, z) in cylindrical or spherical coordinates over the solid shown.Use spherical coordinates. 21. Evaluate B (x2+y2 + z2)2 dV, where B is the ball with center the origin and radius 5.Use spherical coordinates. 22. Evaluate E y2z2 dV, where E lies above the cone = /3 and below the sphere = 1.Use spherical coordinates. 23. Evaluate E (x2 + y2) dV, where E lies between the spheres x2 + y2 + z2 = 4 and x2 + y2 + z2 = 9.Use spherical coordinates. 24. Evaluate E y2 dV, where E is the solid hemisphere x2 + y2 + z2 9, y 0.Use spherical coordinates. 25. Evaluate E xe x2 + y2 + z2 dV, where E is the position of the unit ball x2 + y2 + z2 1 that lies in the first octant.Use spherical coordinates. 26. Evaluate E x2+y2+z2dV, where E lies above the cone z = x2+y2and between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4.Use spherical coordinates. 27. Find the volume of the part of the ball a that lies between the cones = /6 and = /3.Use spherical coordinates. 28. Find the average distance from a point in a ball of radius a to its center.Use spherical coordinates. 29. (a) Find the volume of the solid that lies above the cone = /3 and below the sphere = 4 cos . (b) Find the centroid of the solid in part (a).Use spherical coordinates. 30. 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.Use spherical coordinates. 31. (a) Find the centroid of the solid in Example 4. (Assume constant density K.) (b) Find the moment of inertia about the z-axis for this solid.Use spherical coordinates. 32. 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.Use spherical coordinates. 33. (a) Find the centroid of a solid homogeneous hemisphere of radius a. (b) Find the moment of inertia of the in part (a) about a diameter of its base.Use spherical coordinates. 34. Find the mass and center of mass of a solid hemisphere of radius a if the density at any point is proportional to its distance from the base.Use cylindrical or spherical coordinates, whichever seems more appropriate. 35. Find the volume and centroid of the solid E that lies above the cone z = x2+y2and below the sphere x2 + y2 + z2 = 1.Use cylindrical or spherical coordinates, whichever seems more appropriate. 36. Find the volume of the smaller wedge cut from a sphere of radius a by two planes that intersect along a diameter at an angle of /6.37E 38E Evaluate the integral by changing to spherical coordinates. 41.0101x2x2+y22x2y2 xy dz dy dx Evaluate the integral by changing to spherical coordinates. 42.aaa2y2a2y2a2x2y2a2x2y2 (x2z + y2z + z3) dz dx dy Evaluate the integral by changing to spherical coordinates. 43.224x24x224x2y22+4x2y2 (xz + yz + z2)3/2 dz dy dx A model for the density of the earths atmosphere near its surface is = 619.09 - 0.000097 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 6.370 106 6.375 106. Use this model to estimate the mass of the atmosphere between the ground and an altitude of 5 km.45E 46E Show that x2+y2+z2e-(x2+y2+z2) dx dy dz = 2 (The improper triple integral is defined as the limit of a triple integral over a solid sphere as the radius of the sphere increases indefinitely.)(a) 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 = r cot 0 (or = 0), where 0 0 /2, is V = 2a3.3 (1 cos0) (b) Deduce that the volume of the spherical wedge given by 1 2, 1 2, 1 2 is V = 23133(cos1 cos 2)(2 1) (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, = 2 1, = 2 2, and = 2 1.Find the Jacobian of the transformation. 1. x = 2u + v, y = 4u v Find the Jacobian of the transformation. 2. x = u2 + uv, y = uv2 3E Find the Jacobian of the transformation. 4. x = peq, y = qep Find the Jacobian of the transformation. 5. x = uv, y = vw, z = wu Find the Jacobian of the transformation. 6. x = u + vw, y = v + wu, z = w + uv Find the image of the set S under the given transformation. 7. S = {(u, v)|0 u 3, 0 v 2}; x = 2u + 3v, y = u v Find the image of the set S under the given transformation. 8. S is the square bounded by the lines u = 0, u = 1, v = 0, v = 1; x = v, y = u(1 + v2)Find the image of the set S under the given transformation. 9. S is the triangular region with vertices (0, 0), (1, 1), (0, 1); x = u2, y = v Find the image of the set S under the given transformation. 10. S is the disk given by u2 + v2 1; x = au, y = bv 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. 11. R is bounded by y = 2x 1, y = 2x + 1, y = 1 x, y = 3 x 12E 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. 13. R is lies between the circles x2 + y2 = 1 and x2 + y2 = 2 in the first quadrant 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. 14. R is bounded by the hyperbolas y = 1/x, y = 4/x and the lines y = x, y = 4x in the first quadrant Use the given transformation to evaluate the integral. 15. R (x 3y) dA, where R is the triangular region with vertices (0, 0), (2, 1), and (1,2); x = 2u + v, y = u + 2v Use the given transformation to evaluate the integral. 16. 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(v 3u)Use the given transformation to evaluate the integral. 17. R x2 dA, where R is the region bounded by the ellipse 9x2 + 4y2 = 36; x = 2u, y = 3v Use the given transformation to evaluate the integral. 18. R (x2 xy + y2) dA, where R is the region bounded by the ellipse x2 - xy + y2 = 2; x = 2u - 2/3v, y = 2u + 2/3v Use the given transformation to evaluate the integral. 19. R xy dA, 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 = v 20E (a) Evaluate E dV, 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 = 6378 km and c = 6356 km. 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 0 a b and 0 c d. Compute the work done by determining the area of R.Evaluate the integral by making an appropriate change of variables. 23. Rx2y3xydA. Where R is the parallelogram enclosed by the lines x 2y = 0, x 2y = 4, 3x y = 1, and 3x y = 8 Evaluate the integral by making an appropriate change of variables. 24. R (x + y)ex2 + y2dA. Where R is the rectangle enclose by the lines x y = 0, x y = 2, x + y = 0, and x + y = 3 Evaluate the integral by making an appropriate change of variables. 25. Rcos() dA, whereRis the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)Evaluate the integral by making an appropriate change of variables. 26.Rsin(9x2+4y2) dA,whereRis the region in the first quadrant hounded by the ellipse9x2+ 4y2= 1 Evaluate the integral by making an appropriate change of variables. 27. R ex + y dA,whereRis given by the inequality |x| + |y| 1 Let f be continuous oil [0, 1] and letRbe the triangular region with vertices (0, 0), (1, 0), and (0, 1). Show that Rf(x + y) dA = 01uf(u)du 1RCC 2RCC How do you change from rectangular coordinates to polar coordinates in a double integral? Why would you want to make the change?If a lamina occupies a plane region D and has density function (x, y), write expressions for each of the following in terms of double integrals. (a) The mass (b) The moments about the axes (c) The center of mass (d) The moments of inertia about the axes and the origin 5RCC 6RCC 7RCC 8RCC 9RCC 10RCC 1RQ 2RQ 3RQ 4RQ 5RQ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1401(x2+y)sin(x2y2)dxdy9 Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If D is the disk given by x2 + y2 4, then D4x2y2dA=163 8RQ 9RQ A contour map is shown for a function f on the square R = [0, 3] [0, 31. 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 comers of the squares.Use the Midpoint Rule to estimate the integral in Exercise 1. 1. A contour map is shown for a function f on the square R = [0, 3] [0, 31. 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 comers of the squares.Calculate the iterated integral. 3. 1202(y+2xey)dxdy Calculate the iterated integral. 4. 0101yexydxdy Calculate the iterated integral. 5. 010xcos(x2)dydx Calculate the iterated integral. 6. 01xex3xy2dydx Calculate the iterated integral. 7. 00101y2ysinxdzdydx Calculate the iterated integral. 8. 010yx16xyzdzdxdy Write Rf(x,y)dA as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R.Write Rf(x,y)dA as an iterated integral, where R is the region shown and f is an arbitrary continuous function on R.The cylindrical coordinates of a point are (23,3, 2). Find the rectangular and spherical coordinates of the point.12RE The spherical coordinates of a point are (8, /4, /6). Find the rectangular and cylindrical coordinates of the point.Identify the surfaces whose equations are given. (a) = /4 (b) = /4 Write the equation in cylindrical coordinates and in spherical coordinates. (a) x2 + y2 + z2 = 4 (b) x2 + y2 = 4 16RE Describe the region whose area is given by the integral 0/20sin2rdrd Describe the solid whose volume is given by the integral 0/20/2122sinddd and evaluate the integral.Calculate the iterated integral by first reversing the order of integration. 01x1cos(y2)dydx Calculate the iterated integral by first reversing the order of integration. 01y1yex2x3dxdy Calculate the value of the multiple integral. 21. RyexydA, where R = {(x, y) | 0 x 2, 0 y 3}Calculate the value of the multiple integral. 22. DxydA, where D = {(x, y) | 0 y 1, y2 x y + 2}Calculate the value of the multiple integral. 23. Dy1+x2dA, where D is bounded by y=x, y = 0, x = 1 Calculate the value of the multiple integral. 24. Dy1+x2dA, where D is the triangular region with vertices (0, 0), (1, 1), and (0, 1)Calculate the value of the multiple integral. 25. DydA, where D is the region in the first quadrant bounded by the parabolas x = y2 and x = 8 y2 Calculate the value of the multiple integral. 26. 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 = 2 Calculate the value of the multiple integral. 27. D(x2+y2)3/2dA,where /9 is the region in the first quadrant bounded by the lines y = 0 and y=3x and the circle x2 + y2 = 9 Calculate the value of the multiple integral. 28. DxdA, where D is the region in the first quadrant that lies between the circles x2 + y2 = 1 and x2 + y2 = 2 Calculate the value of the multiple integral. 29. ExydV, where E = {(x, y, z) | 0 x 3, 0 y x, 0 z x + y 30RE Calculate the value of the multiple integral. 31. Ey2z2dV, where E is bounded by the paraboloid x = 1 y2 z2 and the plane x = 0 Calculate the value of the multiple integral. 32. EzdV, where E is bounded by the planes y = 0, z = 0, x + y = 2 and the cylinder y2 + z2 = 1 in the first octant Calculate the value of the multiple integral. 33. EyzdV, where E lies above the plane z = 0, below the plane z = y, and inside the cylinder x2 + y2 = 4 34RE 35RE 36RE 37RE 38RE 39RE 40RE Consider a lamina that occupies the region D bounded by the parabola x = 1 y2 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.A lamina occupies the part of the disk x2 + y2 a2 that lies in the first quadrant. (a) Find the centroid of the lamina. (b) Find the center of mass of the lamina if the density function is (x, y) = xy2.(a) Find the centroid of a solid right circular cone with height hand base radius a. (Place the cone so that its base 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 = 2.45RE Use polar coordinates to evaluate 039x29x2(x3+xy2)dydx Use spherical coordinates to evaluate 2204y24x2y24x2y2y2x2+y2+z2dzdxdy 49RE 51RE A lamp has three bulbs, each of a type with average lifetime 800 hours. If we model the probability of failure of a bulb by an exponential density function with mean 800, find the probability that all three bulbs fail within a total of 1000 hours.53RE 54RE 55RE Use 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.57RE The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point (x0, y0) in D such that Df(x,y)dA=f(x0,y0)A(D) Use the Extreme Value Theorem (14.7.8) and Properly 15.2.11 of integrals to prove this theorem. (Use the proof of the single-variable version in Section 6.5 as a guide.)59RE 60RE 1P Evaluate the integral 0101emaxx2,y2dydxwhere max{x2, y2} means the larger of the numbers x2 and y2.3P 4P The double integral 010111xydxdyis an improper integral and could be defined as the limit of double integrals over the rectangle [0, t] [0, t] as t 1. But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that010111xydxdy=n11n2 Leonhard Euler was able to find the exact sum of the series in Problem 5. In 1736 he proved thatn11n2=26 In this problem we ask you to prove this fact by evaluating the double integral in Problem 5. Start by making the change of variablesx=uv2y=u+v2 This gives a rotation about the origin through the angle /4. You will need to sketch the corresponding region in the uv-plane. [Hint: If, in evaluating the integral, you encounter either of the expressions (1 sin )/cos or (cos )/(1 + sin ), you might like to use the identity cos = sin((/2) ) and the corresponding identity for sin .]7P 8P (a) Show that when Laplaces equation 2ux2+2uy2+2uz2=0is written in cylindrical coordinates, it becomes 2ur2+1rur+1r22u2+2uz2=0 (b) Show that when Laplaces equation is written in spherical coordinates, it becomes 2u2+2u+cot2u+12sin22u2=0 (a) A lamina has constant density and takes the shape of a disk with center the origin and radius R. Use Newtons Law of Gravitation (see Section 13.4) to show that the magnitude of the force of attraction that the lamina exerts on a body with mass m located at the point (0, 0, d) on the positive z-axis is F=2Gmd(1d1R2+d2) [Hint: Divide the disk as in Figure 15.3.4 and first compute the vertical component of the force exerted by the polar subrectangle Rij.] (b) Show that the magnitude of the force of attraction of a lamina with density that occupies an entire plane on an object with mass m located at a distance d from the plane is F=2Gm Notice that this expression does not depend on d.If f is continuous, show that 0x0y0zf(t)dtdzdy=120x(xt)2f(t)dt Evaluate limnn2i=1nj=1n21n2+ni+j.The plane xa+yb+zc=1a0,b0,c0cuts the solid ellipsoid x2a2+y2b2+z2c21 into two pieces. Find the volume of the smaller piece.Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 1. F(x, y) = 0.3 i 0.4 j Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 2. F(x, y) = 12x i + y j Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 3. F(x, y) = 12 i + (y x) j Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 4. F(x, y) = y i + (x + y) j Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 5. F(x,y)=yi+xjx2+y2 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 6. F(x,y)=yixjx2+y2 Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 7. F(x, y, z) = i Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 8. F(x, y, z) = z i Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 9. F(x, y, z) = y i Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. 10. F(x, y, z) = i + k Match the vector fields F with the plots labeled I-IV. Give reasons for your choices. 11. F(x, y) = x, y Match the vector fields F with the plots labeled I-IV. Give reasons for your choices. 12. F(x, y) = y, x y Match the vector fields F with the plots labeled I-IV. Give reasons for your choices. 13. F(x, y) = y, y + 2 Match the vector fields F with the plots labeled I-IV. Give reasons for your choices. 14. F(x, y) = cos(x + y), x Match the vector fields F on 3 with the plots labeled I-IV. Give reasons for your choices. 15. F(x, y, z) = i + 2 j + 3 k Match the vector fields F on 3 with the plots labeled I-IV. Give reasons for your choices. 16. F(x, y, z) = i + 2 j + z k Match the vector fields F on 3 with the plots labeled I-IV. Give reasons for your choices. 17. F(x, y, z) = x i + y j + 3 k Match the vector fields F on 3 with the plots labeled I-IV. Give reasons for your choices. 18. F(x, y, z) = x i + y j + z k Find the gradient vector field of f. 21. f(x, y) = y sin(xy)Find the gradient vector field of f. 22. f(s, t) = 2s+3t Find the gradient vector field of f. 23. f(x, y, z) = x2+y2+z2 Find the gradient vector field of f. 24. f(x, y, z) = x2 yey/z Find the gradient vector field f of f and sketch it. 25. f(x, y) = 12(x y)2 Find the gradient vector field f of f and sketch it. 26. f(x, y) = 12(x2 y2)Match the functions f with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices. 29. f(x, y) = x2 + y2 Match the functions f with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices. 30. f(x, y) = x(x + y)Match the functions f with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices. 31. f(x, y) = (x + y)2 Match the functions f with the plots of their gradient vector fields labeled I-IV. Give reasons for your choices. 32. f(x, y) = sinx2+y2 A particle moves in a velocity field V(x, y) = x2, x + y2. If it is at position (2, 1) at time t = 3, estimate its location at time t = 3.01.At time t = 1, a particle is located at position (1, 3). If it moves in a velocity field F(x, y) = xy 2, y2 10 find its approximate location at time t = 1.05.The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = x i y j to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = y. Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).(a) Sketch the vector field F(x, y) = i + x j and then sketch some flow lines. What shape do these flow lines appear to have? (b) If parametric equations of the flow lines are x = x(t), y = y(t), what differential equations do these functions satisfy? Deduce that dy/dx = x. (c) If a particle starts at the origin in the velocity field given by F, find an equation of the path it follows.Evaluate the line integral, where C is the given curve. 1. C y ds, C: x = t2, y = 2t, 0 t 3 Evaluate the line integral, where C is the given curve. 2. C (x/y) ds, C: x = t3, y = t4, 1 t 2

Page: [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]