Math 2A - Chapter 12 Test  Fall ’07                 Name________________________________

Show your work for credit.  Write all responses on separate paper.  Do not use a calculator.

1.      Write each area as an iterated integral.  You do not need to compute its value.

a.        where  

b.      The area of the region inside the circle  and outside the circle .

2.      Interpret the integral  in polar form and evaluate.

3.      Find the volume of the region E =  
Use spherical coordinates.

4.      Suppose a mass density function  is distributed over the laminar region bounded by  and  

a.       Find the mass of this laminar region.

b.      Find the x-coordinate of its center of mass.

5.      Evaluate .  It may help to reverse the order of integration.

6.      Find the area of the surface z = x + y² that lies above the triangle with
vertices (0,0), (1,1) and (0,1).

7.      Evaluate  where E is the tetrahedron with vertices (0,0,0), (0,1,0), (1,1,0) and (0,1,1).

8.      Use cylindrical coordinates to evaluate  where E is the region that lies inside the cylinder  and between the planes z = 5  and z = 4.

9.      Use spherical coordinates to evaluate , where E lies between
the spheres ρ = 2 and ρ = 4 and above the cone .

10.  Use the given transformation to evaluate , where R is the region bounded by the
lines y =  x, y = x  2, y = 2x and y =  3  2x, and the T is , .

Math 2A - Chapter 12 Test Solutions  Fall ’07

1.      Write each area as an iterated integral.  You do not need to compute its value.

a.        where
ANS:  This is the area of a unit disc = . 
In polar form,  

In polar coordinates, the integral is much simpler.  Double the region where ,  ,  so the integral is  

2.      Interpret the integral  in polar form and evaluate.
ANS:  Look at the upper bound on the inner integral: .  The far left equation here then describes the upper half of a circle of radius 1 in the xy plane, centered at (1,0), and since x ranges from 0 to 2, we get the whole upper half circle, which involves .  The integrand is just r.  Thus  

3.      Find the volume of the region E =  
Use spherical coordinates. 
ANS:  This is the volume inside the sphere ρ = 3 and above the cone .  Thus the volume is  

4.      Suppose a mass density function  is distributed over the laminar region bounded by  and  

a.       Find the mass of this laminar region.
ANS:  First get a picture of the region.  I had a hard time getting Maple to do this.  I ended up with the following command:
> with(plots):
 implicitplot((x+2*y^2-3)*(x-y^2), x = -0.5 .. 3, y = -1 .. 1, filledregions = true, resolution = 100000, coloring = [gray, white]);

 
Thus the center of mass is .

ANS:   is a parameterization of the surface, so the area is   

7.      Evaluate  with E the tetrahedron having vertices (0,0,0), (0,1,0), (1,1,0) and (0,1,1).
ANS: The four planes that contain the tetrahedron are the xy-plane ((0,0,0),(0,1,0),(1,1,0)) the yz-plane ((0,0,0),(0,1,1),(0,1,0)), the plane y = 1: ((0,1,0),(1,1,0),(0,1,1)), and the plane
z = y  x:  ((0,0,0),(1,1,0),(0,1,1)).  Integrating out the z first, we have  

8.      Use cylindrical coordinates to evaluate  where E is the region that lies inside the cylinder  and between the planes z = 5  and z = 4.
ANS: 

9.      Use spherical coordinates to evaluate , where E lies between
the spheres ρ = 2 and ρ = 4 and above the cone .
ANS:   

Corners of a rectangular region in the uv-plane.  Thus the integral is