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Unformatted text preview: [5453— F0/ MATHEMATICS 182 TEST IIIA 1) F = (y sin z)i + (a; sin z)j + (my cos z)k. 2) Find the mass of a thin wire lying along the curve r(t) = x/i ti+ x/i tj + (4 —t2)k, 0 S
t» g 1, if the density is 6 = 3t. 3) Find the mass of a thin plate covering the region outside the circle r = 3 and inside
the circle 7' = 6 sin0 if the plate’s density function is 6(x, y) = 1/7". 4) Set up but do not evaluate integrals for the following: a) The solid bounded below by the hemisphere p = 1, z 2 D, and above by the
cardioid of revolution p = 1 + cos c5. b) Find the volume of the region bounded above by the paraboloid z = 9 — x2 — y2, below by the my—plane, and lying outside the cylinder 3:2 + y2 = 1. c) Find the moment of inertia of a right circular cone of base radius a and height
h about its axis. (Hint: Place the cone with its vertex at the origin and its axis
along the zaxis.) 5) Find the circulation and ﬂux of the ﬁelds F 2 xi + yj xj around and across the
following curve: The ellipse r(t) = (cos t)i + (4 sin t)j, D S t S 27r. 6) F (x, y, z) = xyz over the cube in the ﬁrst octant bounded by the coordinate planes
and the planes at = 2, y = 2, and z = 2. 7) a) Solve the system u = 2:1; — 3y, 1) r: —a: + y for :1: and y in terms of u and ’0. Then
ﬁnd the value of the Jacobian 8(ac, y) / 6(u, 1)). b) Find the image under the transformation u = 2:1: — 3y, '0 : ~56 + y of the parallelogram R in the wyplane with boundaries at: = ~3, x = 0, y = :13, and y = as + 1. Sketch the transformed region in the uv—plane. )Use the transformation and parallelogram R 1n a) to evaluate the integral C//(2:c— y)d:cdy. MATHEMATICS 182 TEST 3 (10 pts) 1) Find I z, the moment of inertia with respect to the z—axis of the volume between
2 = :82 +312 and z = 1 if the density 6 = z. (10 pts) 2) Change the integral 4— \/——z—2
/02 / 6xdydx
4— m2 to polar coordinates and evaluate it. (10 pts) 3) Find the area outside 7' = 1 and inside 7' = 1 + cos 0. (10 pts) 4) Evaluate the line integral of f (x y, z)— — x + y + 2 along the line connecting (1,1,1)
to (2 3 2.) (10 pts) 5) Find the work done by the force 13" 2 ﬁt + 2% over the curve m=cost y=sint z=t,0<t<7r. (10 pts) 6) Find the volume above 2 = 3/2 and below z = 4 between at = O and x = 1. (30 pts) 7) Set up but do not evaluate integrals for the following. a) The mass of the tetrahedron with corners (0,0,0), (1,0,0), (0,1,0) and (0,0,1)
if the density 6 : 2y, b) The center of mass of the plate bounded by the parabola y2 = 4:1: and the line
a: +31 2 4 if the density 6 z 1, c) The volume between the spheres :62 +312 + z2 = 9 and $2 + y2 + 22 = 1 above the
cone (,0 = 77/4. (10 pts) 8) a) Solve the system
u=ac+2y and vzx—yfor x
y and ; Find the Jacobian J (u, '0). b) Sketch the region in the x — y plane bounded by y = 0, y = x, and a: + 2y = 2.
What is the image in the u — 1) plane? 2/3 2—2y
0) Change / / (a: + 2y)ey_$dxdy into an integral over a domain in the 0 y
u — 1} plane. Do not evaluate the integral. ...
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 Fall '09
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