February 8th , 2016
Time Limit: 50 Minutes
This exam contains 9 pages (including this cover page) and 6 problems.
You may not use your books, notes, or any calculator on this exam. All you should have with
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Assignment Homework1 due 01/14/2016 at 11:59pm AST
1. (1 point) Evaluate the triple integral
where E is the solid: 0 z 6, 0 y z, 0 x y.
1x2 y2 +1
1x2 y2 1
2x2 x2 +y2
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Assignment 5: Dierential Equations: Solutions
1. (a) The auxiliary equation is x2 2x + 4 = 0. Using quadratic
formula, one can nd that the roots are x = 1 i 3. Thus the
general solution is
y = ex (c1 cos 3x + c2 sin 3x)
(b) The complementary equation
Some review questions
1. Evaluate D ex+y dA where D is a region bounded by lines x = 0, y = 0 and y + x = 1.
(x2 + y 2 + z 2 ) dV where D is a region bounded by the cylinders x2 +
y 2 = 1 and x2 + y 2 = 4 with 0 z 1 and with x,
Some additional review questions
See also midterm review sheet, go over all quizzes and midterm, and suggested
problems. All quizzes/midterm/reviews will be posted to the course website,
1. State the Diverge
You have 30 minutes
1. Let F (x, y, z ) = (z 2 x, 1 y 3 + tan z, x2 z + y 2 ).
(a) Use the divergence theorem to compute the ux S F n dS where S is surface of
a unit sphere x + y + z = 1, and n is outwards-pointing unit norma
You have 20 minutes
1. Compute C F dx where F = (x2 + y, 4x y 2) and C is the positively oriented boundary
curve of a region D that has area 4.
2. For each of the vector eld F below, determine whether it is conservative or not. If it
You have 25 minutes
1. Determine the volume of the solid obtained by intersecting the unit sphere and a cone
whose vertex is at the origin and which opens up 45 degrees (as measured from its axis
to its side; see board).
2. By making
You have 30 minutes
1. (a) Find the general solution to the ODE
y 4y + 5y = 0
(b) Find the general solution to the ODE
y 4y + 5y = e2x .
2. Use the method of variation of parameters to nd a particular solution to the ODE
y + y =
Math 2002 Midterm
1. Let E be the solid which is bounded by the cylinder x2 + y 2 = 1, lies above the parabola
z = x2 + y 2 and below the plane z = 2 + y. Determine the volume of this solid.
2. Compute the surface integral I =
S : x2 + y 2 + z 2 = 1.