Calculus 3       Hour Exam 3   - Math 241    Spring, 2010

 

 

 

 

 

1.

 

 

Sketch the region of integration and then reverse the order of integration

 

                          1       x-1

 for                    ∫       ∫      f(x,y)  dy dx

                         0   - √ 1 - x2

 

Answer:

                      0       √ 1 - y2

                      ∫       ∫      f(x,y)  dx dy

                    -1       y + 1

 

 

 

2.

 

Find the mass of the lamina bounded by the circle  ( x – 1)2  +  y2  =  1 with

density     δ( x,y )  =  1/x.       Use polar coordinates.

 

Hint:  Draw sketch of region.      Answer:  2 π

 

 

 

 

 

 

 

 

3.

 

 

Let  E  be the solid in the first octant bounded by the surfaces  x = 0,  z = 0,   y = 2x,

and  y2  + 4z2  =  4.  Find the limits of integration in the following integral:

 

                                           ?     ?    ?

        ∫   ∫   ∫  f dV  =            ∫     ∫     ∫     f dy dz dx

                 E                       ?     ?    ?

 

Sketch the solid and its projection to the xz – plane.

 

 

                1     √(1-x2 )    √(4 – 4z2 )

Answer:  ∫        ∫              ∫                      f dy dz dx

                0       0            2x

 

 

 

 

4.

 

Let  E  be the solid above the cone   z =  √(3x2  +  3y2 )  and inside the sphere

 

x2  +  y2  +  (z – 1)2  =  1 .  Evaluate  ∫   ∫   ∫E  z-1  dV   using spherical coordinates.

 

Answer:  π / 2

 

 

 

 

 

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