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 - >) 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