Calculus 3       Hour Exam 3   -  Math 241    Fall, 2010

 

 

1a.

 

 

State, with hypotheses, Green’s Theorem.

 

Answer: 

 

 

 

 

1b.

 

Compute   ∫ xy dx  + xy dy    using an integral along the path, C,  where C is along the

x-axis from  (-1,0)  to  (1,0)  followed by a semicircular path (or radius 1) from (1,0)

returning to (-1,0).  Hint:  Draw this path.

 

   Answer:  2/3

 

 

1c.

 

 

Compute   ∫ xy dx  + xy dy    using Green’s Theorem.              Answer:    2/3

 

 

2a.

 

Let  R  be the region between  x  =  √( 1 – x2– z2 )  and  x   = √( 4 – x2– z2 )  , such

that   x  ≥  1 .    Draw the region.

 

 

 

 

 

2b.

 

State the transformation formulas for spherical and cylindrical coordinates.

 

                       Spherical                         Cylindrical

Answers:      z  =  ρ cos φ                        z  =  z

                    x  =  ρ sin φ cos θ                x  =  r cos θ

                    y  =  ρ cos φ cos θ               y  =  r sin θ

 

 

 

 

 

2c.

 

Compute   ∫  ∫  ∫  y2  dx dy dz  using one of the coordinate systems from part 2b.

                       R

 

where R is the region between  x  =  √( 1 – x2– z2 )  and  x   = √( 4 – x2– z2 )  , such

that   x  ≥  1 .

 

Answer:    53π / 60

 

 

 

 

 

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