{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

182E3-S2001

# 182E3-S2001 - [5453-— F0 MATHEMATICS 182 TEST IIIA 1 F...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 z-axis.) 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 wy-plane 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. ...
View Full Document

{[ snackBarMessage ]}