Directions: Do not simplify unless indicated. No calculators are permitted. Show all work as appropriate for the methods taught in this course. Partial credit will be given for any work, words or...

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Directions: Do not simplify unless indicated. No calculators are permitted. Show all work as
appropriate for the methods taught in this course. Partial credit will be given for any work, words
or ideas which are relevant to the problem.
Important: Out of all the integrals that ask to be evaluated only one of them actually requires
integration, the rest use theorems which make them fast.
1. Consider the following three pictures of vector fields. Each axis goes from −3 to 3 in each of [20 pts]
the x and y directions and the vectors are anchored at integer coordinates.
(I) (II) (III)
For each of the following (a), (b), and (c), plug in a single point (x, y) which allows you to
identify which of the vector fields (I), (II), or (III) matches the vector field.
(a) F(x, y) = 0.3y i− 0.3x j
(b) F(x, y) = 0.3y i + 0.3x j
(c) F(x, y) = 0.3x i + 0.3 j
2. Parts (a) and (b) are unrelated.
(a) Let Σ be the sphere with equation x2 + y2 + z2 = 9 with inwards orientation. Evaluate: [8 pts]∫∫
Σ
(x i + 2y j + 3z k) · n dS
(b) Suppose C is the curve parametrized by r(t) = t2 i + t j + 2 k for 0 ≤ t ≤ 2. Evaluate: [12 pts]∫
C
y dx + z dy + z dz
3. Suppose Σ is the portion of the parabolic sheet y = 4−x2 in the first octant and below z = 3. [20 pts]
The mass density at each point is given by f(x, y, z) = x + y + z. Set up an iterated integral
for the mass of Σ.
Do Not Evaluate This Integral.
4. Parts (a) and (b) are unrelated.
(a) Let C be the clockwise triangle with vertices (5, 0), (5, 10), and (0, 10). Evaluate: [10 pts]∫
C
(x + 2y) dx + (4x + y) dy
(b) Let C be the straight-line segment from (1, 2,−2) to (8, 0, 5). Evaluate: [10 pts]∫
C
(y i + x j + 1 k) · dr
5. Suppose C is the intersection of the plane 2x + y = 10 with the cylinder x2 + z2 = 9 with [20 pts]
counterclockwise orientation when viewed towards the origin from the positive y-axis. Apply
Stokes’ Theorem to the integral∫
C
(x + y) dx + y2 dy + (xyz) dz
and proceed until you have an iterated double integral.
Do Not Evaluate This Integral.
Answered Same DayMay 10, 2021

Solution

Rajeswari R answered on May 11 2021
24 Votes

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