Calculus III - Exam 5 - Fall 2002

 

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1.  Calculate the line integral, , with   ,  over the elliptical path g  parametrized by        from    t = 0   to   t = p.

(A)  p/2 + 3             (B) p + 4             (C) 2p + 5
     
(D) 3p + 6             (E) 4p + 7

2.   Calculate the line integral, , with   ,  over the parabolic path g  parametrized by     from   t = 0   to   t = 1.

(A) 53/5       (B) 54/7       (C) 55/7       (D) 56/9      (E) 57/10

 3.   Calculate   with   ,  where   g(x, y) = 5x3 – 2xy2,   over the path  g   parametrized by       from   t = 0   to   t = 1.

(A) 1          (B) 0        (C) –1         (D) –2        (E) –3

 4.   Suppose that    with  g(0,0) = 0.      Then   g(x, y) = ?

(A)  x sin y             (B) x2 cos y             (C) x2 sin y

(D) x3 cos y             (E) x3 sin y

 5.  Calculate      with    ,   over the path g  from the point (1, p) to  (2, p/2)  along a straight line.    

        

(A)  1             (B) 2              (C) 3              (D) 4             (E) 5

6.    Suppose that    with  g(0,0) = 0.     Then   g(x, y) = ?

(A)  x2y + xy2           (B) x2y – xy2         (C) xy – x2y2           

(D) x2y2 – xy           (E) x2y2 + xy

7.   Calculate    with    ,   over the path g  from the point ( 3, 1 ) to ( 5, 2 ) along the line segment connecting these two points.

 

(A)  21             (B) 22              (C) 23              (D) 24             (E) 25

8.   Calculate   with    ,   over the path g  from the point (p, –1)  to (0, 1)  along the curve    y = cos x.

(A)  p2 +p            (B) p2 – p           (C) p2 +2p             (D) p2 – 2p             (E) p3 +p2

 9.  If    and  g  is the boundary of the square with
      vertices  (1,1), (3,1), (3,3), (1,3),  then  

(A)  1       (B) 0       (C) –1       (D) –2       (E) –3

 10.  Suppose    and region R has an area of  7  and has boundary g  that is a simple closed curve.        Then  

(A)  0       (B) –18       (C) –19       (D) ­–20       (E) –21

 11.  Suppose , where   g(x,y,z) =  x2 cos y – sin(yz).    If  g  is the ellipse parametrized by  

(A) 1       (B) 0       (C) –1       (D) –2        (E) –3

12.  If  g  is the circle   x2 + y2 = 4, then   =  ?

(A) 0         (B) 4p        (C)  p2 + 2 

(D)  p2 + 3        (E) 2p2

13.  If    and  g  is the unit circle centered at the origin.  If R is the circular interior of  g,  then     =  ?      [ Hint:  this time it’s easier to compute the line integral! ]

(A) 0       (B) p       (C) –p       (D) p/2       (E) –p2/2

14.  Compute the (upward) flux of the field  through the portion of the
 paraboloid surface    z = 1 – x2 – y2   that lies above (and/or below) the rectangle in the xy-plane with corners at (0,0), (1,0), (1,1), (0,1).

                         (A)  0             (B) 37/16             (C) 35/16             (D) 7/3             (E) 8/3            

15.  Compute the divergence of the vector field  .

(A)  0                (B) 2–x+xy             (C) 1+x–2xy
 
(D) 1+2x+xy             (E) 1+x+xy

16.  Compute the outward flux of through the surface of the cube defined by .

                         (A)  0        (B) 7/3        (C) 5/4        (D) 7/4       (E) 8/3            

17.   Compute the curl of the vector field .

(A)  (0,0,0)             (B) (–xz, yz, x)         (C) (xz, –yz, y)

(D) (xy, –xz, z)              (E) (–xz, xz, x)

18.   Compute the circulation of around the rectangular path g  from (0,0,0) to (1,0,0) to (1,1,1) to (0,1,1) and then back to (0,0,0).            

(A) 0       (B) –8/17       (C) –7/15       (D) –6/13       (E) –5/12

 19.   Suppose  g(x,y,z) = xy2z3.   Then  

(A)  (0,0,0)             (B) (–xz, yz, x)             (C) (xz, –yz, y)

(D) (xy, –xz, z)              (E) (–xz, xz, x)

 20.  Compute the circulation of the field (featured in #19),  around the rectangular path g  from (0,0,0)  to (1,0,0) to (1,1,1) to (0,1,1) and then back to (0,0,0).             

 (A) 0       (B) 7/3       (C) 5/4       (D) 7/4       (E) 8/3