190 CHAPTER 2 . MULTIPLE IN 50. Find the has a constant density. xy dA , where R is the region bounded by the hyperbolas 51. Find the average square...
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# Stuck on problems 43 and 44, need help in preparation for an exam please. src="/qa/attachment/11094932/" alt="256E1897-2190-4417-BB5F-37D9F26473AA.jpeg" /> ATTACHMENT PREVIEW Download attachment 256E1897-2190-4417-BB5F-37D9F26473AA.jpeg 190 CHAPTER 2 . MULTIPLE IN 50. Find the has a constant density. xy dA , where R is the region bounded by the hyperbolas 51. Find the average square of 35 . the origin. xy = 1 and xy = 4, and the lines y = 1 and y = 3 52. Find the average distance b 36. (/ ( x - &gt;) Vx - 2y dA, where R is the triangular region and the x-axis. R 53-56. Ellipsoid problems Let bounded by y = 0, x - 2y = 0, and x - y = 1 *2 / a2 + y2 / b2 + 22/ c2 = 1, w 37-40. Jacobians in three variables Evaluate the Jacobians J(u, v, w) real numbers. Let T be the trans for the following transformations. 53. Find the volume of D. 37. x = v+ w,y = u+ w,z = utv 38. x = utv- w,y = u-v+ w,z= -utv+w 54. Evaluate / Ixxzl d4. 39. x = vw, y = uw, z = u2 - v2 55. Find the center of mass of th 40. u = x - y, v = x - z, w= y+ z (Solve for x, y, and z first.) it has a constant density. 41-44. Triple integrals Use a change of variables to evaluate the 56. Find the average square of t following integrals. the origin. 41. xy dV; D is bounded by the planes y - x = 0, 57. Parabolic coordinates Let D y = 2uv. y - x = 2, z - y = 0, z - y = 1, z = 0, and z = 3. a. Show that the lines u = 42. / dv; D is bounded by the planes y - 2x = 0, y - 2x = 1, the xy-plane that open in on the positive x-axis. z - 3y = 0, z - 3y = 1, z - 4x = 0, and z - 4x = 3. b. Show that the lines v = the xy-plane that open in 43. z dv; D is bounded by the paraboloid z = 16 - x2 - 4y? on the negative x-axis. c. Evaluate J(u, v). and the xy-plane. Use x = 4u cos v, y = 2u sin v, z = w. d. Use a change of variables bounded by x = 4 - yz 44. // av; D is bounded by the upper half of the ellipsoid e. Use a change of variables rectangle above the x-ax x2/9 + y2/4 + z? = 1 and the xy-plane. Use x = 3u, x = 9 - y2/36, x= y2/ y = 2v,z = W. f. Describe the effect of the y = u2 - v2 on horizont Further Explorations 45. Explain why or why not Determine whether the following state- Applications ments are true and give an explanation or counterexample. 58. Shear transformations in R a. If the transformation T: x = g(u, v), y = h(u, v) is linear in u by x = au + bv, y = cv, w and v, then the Jacobian is a constant. numbers, is a shear transfor b. The transformation x = au + by, y = cu + dv generally { (u, v): 0 Sus1, Osv maps triangular regions to triangular regions. of S. c. The transformation x = 2v, y = -2u maps circles to circles. a. Explain with pictures the 46. Cylindrical coordinates Evaluate the Jacobian for the trans- formation from cylindrical coordinates (r, 0, Z) to rectangular b. Compute the Jacobian of coordinates (x, y, z): x = r cos 0, y = r sin 0, z = Z. Show that c. Find the area of R and co J(r, 0, Z) = r. d. Assuming a constant dens terms of a

43. The given integral will... View the full answer

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