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Name:
Section (circle one):
AD1 AD2 ADA ADB ADC
ADD ADE ADF ADG ADH
ADI ADJ ADK ADL ADM
BDA BDB BDC BDD BDE
BDF BDG BDH BDI BDJ
BDK BDL BDM BDN BDO
BDP BDQ
READ ALL INSTRUCTIONS CAREFULLY. Write legibly, and use the boxes for your final answers where
provided.
Be sure to use correct notation; in particular, distinguish vectors from scalars by arrow notation, use explicit
clearly visible dots for dot products, etc.
An answer alone, without justification, will not earn full credit (with the exception of the multiple choice
Scantron problems 1, 4, 5, and 7). If you make a mistake, cross out all of your incorrect work. We will
take points off for incorrect work that is not crossed out, even if the correct answer is given elsewhere.
Problem Point Value Test Score
1 5 Scantron
4 4 Scantron
5 6 Scantron
7 3 Scantron
Total 40
- (5 points) Fill in the blanks in the integrals below that compute the length L of the curve C given by
the top half of the ellipse x
2
2 = 4 (that is, the part of the ellipse that lies on or above the x-axis
and connects the two points (2; 0) and (−2; 0)). Use the parametrization r (t) = ⟨ 2 cos t; sin t⟩ for the
ellipse. You do NOT have to evaluate the integral. Hint: Sketch the ellipse and the curve C.
L =
C
Line
1 ds^ =
Second number in line
First number in line
Line
3 dt
Pencil your answers into the corresponding lines in your Scantron bubble sheet.
Line
A = x
2
2
B = cos
2 t + 4 sin
2
t
C = 1
D = F ( r (t)) r
′ (t)
E = 4
Line
A = 0 to
B = 0 to 2
C = 0 to 1
D = −
to
E = −
to
Line
A = 2
B = − 2 sin t + cos t
C =
p
− 2 sin t + cos t
D =
1 + 3 sin
2 t
E = 1 + 3 sin
2
t
- (5 points) Evaluate the line integral of the vector field
F (x; y) = ⟨−y; x⟩
along the curve C given by the arc of the circle of radius 2 from (0; − 2 ) to (0; 2). (Of course there are
two such arcs; C is the one that lies on and to the right of the y-axis.) Show your work.
C
F d r =
- (2 points each = 4 points) Below are the plots of two vector fields F and the graphs of curves C
superimposed on each plot. For both figures, determine whether the line integral of the vector field
along the curve is:
A = positive or
B = negative or
C = zero
and pencil your answers into the corresponding lines in your Scantron bubble sheet. (Options D and
E are not valid answers in this problem.)
The figure on the LEFT corresponds to line
4 , and the figure on the RIGHT corresponds to line
in your Scantron bubble sheet.
- (2 + 2 + 1 + 1 points = 6 points) Match the plots of the vector fields F in the above figures with
their equations, and pencil your answers into the corresponding lines in your Scantron bubble sheet.
(Options D and E are not valid answers in this problem.)
In line
6 of your Scantron bubble sheet, mark whether the vector field on the LEFT is:
A F (x; y) = y⃗ { − ⃗ȷ or
B F (x; y) = x⃗ { + y⃗ ȷ or
C F (x; y) = xy⃗ { + (x − y)⃗ȷ
In line
7 of your Scantron bubble sheet, mark whether the vector field on the RIGHT is:
A F (x; y) = 0:5y⃗ { or
B F (x; y) = ⃗{ or
C F (x; y) = 0:5x⃗ ȷ
Also match the two curves C in the same figures with their vector equations, and pencil your answers
into the corresponding lines in your Scantron bubble sheet.
In line
8 of your Scantron form, mark whether the curve on the LEFT is given by the vector equation:
A r (t) = ⟨ 4 cos t; 4 sin t⟩; 0 t or
B r (t) = ⟨t; 4 cos t + 4 sin t⟩; 0 t =2 or
C r (t) = ⟨ 4 cos t; 4 sin t⟩; 0 t =
In line
9 of your Scantron form, mark whether the curve on the RIGHT is given by the vector equation:
A r (t) = ⟨t; t
2 ⟩; 0 t 5 or
B r (t) = ⟨t; −t⟩; 0 t 5 or
C r (t) = ⟨t; t⟩; 0 t 5
- (7 points) Evaluate the double integral ∫∫
D
3y dA;
where D is the region bounded by the parabola y
2 = x and the line x − 2y = 0.
First sketch the region D. Then set up an iterated integral, and evaluate it. Show your work.
D
3y dA =
