CALCULUS 265 Honors                        Student Name_____________________
TEST 2 September 29, 2000
Instructor: Hentzel


1.  Find the volume above z = x^2+y^2 which is below the plane  z = 16;





















      





                                  1. Volume = ____________________________

2.  Find the surface area of the surface z = x^2 + y^2 below the plane z = 16.
































                         2. Surface Area = ______________________________

3.  Convert the following into Cylindrical coordinates.  Do not integrate.

       3   +Sqrt[9-x^2]   Sqrt[25-x^2-y^2] 
      INT   INT            INT               Sqrt[x^2+y^2] dz dy dx.
       0   -Sqrt[9-x^2]     0 













theta = ________    r = _________    z = ________

   INT               INT             INT                             dz dr dtheta

theta = _________   r = _________    z = ________  

4.  Convert the following into Spherical coordinates.  Do not integrate.

       3    Sqrt[9-x^2]   Sqrt[9-x^2-y^2]
      INT   INT            INT             x^2+y^2+z^2  dz dy dx
       0     0              0













theta = _____   phi = __________      p = ___________

     INT           INT                 INT                         dp dphi dtheta

theta = _____   phi = __________      p = ___________

5.  Set up the Integral for the moment about the xy plane  of the portion of 

the cone z = 4 Sqrt[x^2 + y^2]  below z = h in Spherical coordinates.  Do not

integrate.











theta = _____   phi = __________      p = ___________

     INT           INT                 INT                         dp dphi dtheta

theta = _____   phi = __________      p = ___________

6.  Set up the Integral for Iz  for the portion of the sphere x^2+y^2+z^2 = a^2

above z = b in cylindrical coordinates.  Do not integrate.















theta = ________    r = _________    z = ________

   INT               INT             INT                             dz dr dtheta

theta = _________   r = _________    z = ________  

7.  Set up the Integral for the surface area of the portion of the sphere

x^2 + y^2 + z^2 = 5^2 between the planes  z = 3 and z=4 using polar coordinates.

Do not integrate.













theta = ________    r = _________    

   INT               INT                                    dz dr dtheta

theta = _________   r = _________     

8.  Set up the integral for the volume of  the "sno-cone" which is the

intersection of the Sphere x^2 + y^2 + z^2 = a^2 and the cone

z^2 = 9(x^2 + y^2) in spherical coordinates.  Do not integrate.











theta = _____   phi = __________      p = ___________

     INT           INT                 INT                         dp dphi dtheta

theta = _____   phi = __________      p = ___________

9.   Set up  the integral for the volume inside the sphere x^2+y^2+z^2 = a^2

and above   z = Sqrt[x^2 + y^2] in Cylindrical coordinates.   Do not integrate. 

























theta = ________    r = _________    z = ________

   INT               INT             INT                             dz dr dtheta

theta = _________   r = _________    z = ________  
     
10.  Give a geometrical interpretation of the triple integral.


   theta= 2 Pi  phi= Pi/2   p=6 Cos[phi]
        INT       INT          INT        p^2 Sin[f] d(rho) d(phi) d(theta)
     theta=0      phi=0        p=0