Calculus III - Exam 4 - Fall 2002

 

Note that problems #2 and #6 have been changed from what they were on the hard-copy.   Also note the extensive hints I've added for problem #7.

 

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1.   Let  D  denote the rectangle 0 < x < 1,  0 < y < 2. 
      Then    

 

(A)           (B)           (C)

 

(D)             (E)

 

 

2.  If  R  denotes the region described by   0 < x < p/2,  ln x < y < 2 ln x, 
      then

 (A)  p / 2                (B)  2p  + 1              (C)  2p  – 1

 

(D)  p  – 3             (E)  p  + 3 

 

 

3.  Let B  be the rectangular solid defined by     
      Then  

(A) 0       (B) 1       (C) 1/2       (D) 1/3       (E) 1/4

 

 

4.  Let  W  be the region  0 < x < 1,   2x < y < 3x,   xy/5 < z < 5xy. 
      Then   

(A) 1098/37              (B)1098/35            (C)1097/37

 

(D)1097/35              (E)1096/37

 

 

5.  If  S  denotes the unit-hemisphere above the xy-plane,
     then 

 

(A)  13p /14                (B)  12p /13              (C)  11p /16

 

(D)  10p /17               (E)  10p /19

 

 

 

 

 

6.   Let  R  denote the region defined by  2 < xy <  5,    1 < y – x < 3. 
      Then 

Hint:  Change coordinates to   u = xy,  v = y – x.

 

(A)  109                     (B)  109.25                   (C)  109.5

 

(D)  109.75                        (E)  110

 

 

 

 

 

7.   Let  R  denote the region   1 < xy <  3,    1 < x2 - y2 < 4.      
      Then 

Hints: 

(i)   Change coordinates to   u = xy,    v = x2 - y2 .  

 

(ii)   Make use of the fact that  (x2 + y2)2  =  4u2 + v2.

 

(iii)  To find the Jacobian determinant , just take the reciprocal of the easy-to-compute Jacobian determinant  ,  using the above "fact" to convert that determinant expression expressed in terms of x and y to one written in terms of u and v.

 

(A)  2.25        (B)  2.5       (C)  2.75       (D)  3       (E)  3.25

 

8.   Change from Cartesian to polar coordinates to compute:  

 

(A)           (B)          (C)

 

(D)            (E)

 

 

9.   Evaluate    over the solid ellipsoid .

Hint:   let  x = au,   y = bv,   z = cw, then integrate over an appropriate region in uvw–space.

 

(A)                (B)           (C)

 

(D)                  (E)

 

 

10.   Let  D  be the unit circle in the  xy–plane.   Then     

(A)            (B) 

 

(C)            (D)

 

(E)

 

11.   Let    define a coordinate transformation.    

        Then    dx dy =       ?       du dv .

 

(A)           (B)           (C)

 

(D)                   (E)

 

 

12.   Find the volume of the finite region enclosed by the two paraboloids,

 

 

(A) 8p         (B) 12p        (C) 16p         (D) 18p       (E) 20p

 

 

13.   Let    define a coordinate transformation.   

        Then   dx dy =       ?       du dv  .

 

(A)                (B)              (C)

 

(D)                   (E)