Calculus 3    Hour Exam 3  (continued)    -  Math 241    Fall, 2010

 

 

 

 

 

3a.

 

 

Parameterize the path from A to B to C.  Let  t  be the parameter where  0 ≤ t ≤ 1.

                                  

 

 

3b.

 

Compute the work done by  F  =  <x2,  -y>  along the path in part 3a.         Answer:  2/3

 

 

3c.

 

 

Is  F  conservative?  Justify.              Answer:    yes

 

 

 

 

 

4a.

 

Suppose  T  is a  C1  map, with nonzero Jacobian, and  T(S)  =  R and is one-to-one

(save possibly on the boundary), where  S  has coordinates  u,v  and  R  has coordinates

x, y  and  T  is described by  x = g(u,v), y = h(u,v).    Express the change of variables

formula

                        ∫  ∫   f(x,y) dx dy  =  ?

 

Answer:    ∫  ∫   f(u,v) J(u,v) du dv     where  J (u,v) is the Jacobian of transformation.

 

 

 

 

 

 

 

 

4b.

 

Let  R  be the region in the  x, y plane between circles of radius 1 and 2, and between

the lines  x = 0  and  x = y.   Evaluate  ∫  ∫   (x2  +  y2) dx dy   for the region  R (below)

using the transformation in 4a.

                             

 

Answer:    15 π / 16

 

 

 

 

 

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