Calculus 3  Hour Exam 2  (continued) -   Math 241     Fall, 2010

 

 

 

4a.

 

 

For a region  R  in three dimensions, write down the formulas for center of mass,

First in terms of m, Myz, Mxz, Mxy, and then in terms of integrals.  Let δ be mass density.

 

Answers:   m = ∫∫∫ δ dV,  xbar = Myz/m  =  ∫∫∫ xδ dV,   ybar = ∫∫∫ yδ dV, zbar = ∫∫∫ zδ dV,

 

 

 

 

4b.

 

For the region in Problem 3a  with  δ(x,y,z) = y, write down appropriate limits and

integrals to compute the center of mass.

 

Answers:  xbar = Myz/m  =  ∫∫∫ xδ dV,   ybar = ∫∫∫ yδ dV/m, zbar = ∫∫∫ zδ dV/m,

 

where  δ = y  and same limits as in Problem 3b.

 

 

 

4c.

 

 

Find the location of the center of mass,  xbar, ybar, and zbar.

 

Answers:  By symmetry  xbar and zbar = 0,  ybar  =  2.49

 

 

 

5a.

 

 

Draw the region  R  in the positive first quadrant (x ≥0, y ≥0, z≥0)  where

x + y  ≤ 1   and   x2  +  y2  +  z2  ≤ 1  Hint:  you’re chopping a slice off a

portion of a sphere.

 

 

 

 

5b.

 

 

Determine appropriate limits to compute (order cannot be changed) the volume i.e.

    ∫ ∫ ∫ dz dy dx

                                            x = 1     y = 1-x    z = √(1 – x2 – y2)

                          Answer:         ∫           ∫             ∫           dz dy dx

                                           x = 0      y = 0       z = 0