MTH 277 Graded Assignment 7 (modules 15 and 16)
As usual, “evaluate” means “by hand, showing all steps and techniques.” The only time you can go pure
Maple for the computation is with the last problem on center of mass. The regions are all familiar
shapes and should be hand-sketchable as well, or you can plot them in one of the many 3D plotting
things we have available.
Problem 1: Evaluate
(a)
(b)
1 1

e2 y e x
0 0 0
2

  
0
4
0
1
dz dx dy
1  ex
cos
0
 2 sin  d  d d
Problem 2: Describe and sketch the solid whose volume is given by the iterated integral shown below
(make a rough sketch of both the 3D soild, and a 2D sketch of its floor in the xy plane). Evaluate the
integral to find the volume.
V 
2
0
3
 
1
3 r 2
0
r dz dr d
Problem 3:
Set up and evaluate the iterated integral in cylindrical coordinates that gives the volume of the solid
bounded by the paraboloids z  3x 2  3 y 2 and z  4  x 2  y 2 .
Problem 4:
Use cylindrical coordinates to evaluate

E
( x3  xy 2 ) dV
where E is the solid in the first octant that lies beneath the paraboloid z  1  x 2  y 2 . Be sure to
include a sketch of the region E .
Problem 5:
Use spherical coordinates to evaluate

D
xe( x
2
 y 2  z 2 )2
dV
where D is the solid that lies between the spheres x 2  y 2  z 2  1 and x 2  y 2  z 2  4 in the first
octant.
Problem 6: For each of the solid regions described below with given density functions  , set up the
integrals for mass m , and the moments M yz , M xz , and M xy in the requested coordinate systems.
Include a sketch of the region – either a hand sketch or a Maple plot is fine. Then, use Maple to compute
the values of the integrals and obtain the coordinates of the centers of mass for each solid region.
Although Maple won’t care about the coordinate system, (and would be happy to do everything set up
in rectangular), please do convert to the specified coordinate system – it’s the converting that I’m
checking.
(a) [Rectangular coordinates] The tetrahedron formed by the plane 3x  2 y  z  6 in the first
octant (so the planes x  0 , y  0 , and z  0 form the other three sides) with density
 ( x, y, z)  x  y 2  z 3 .
(b) [Cylindrical coordinates] The quarter of the cone z 
x 2  y 2 that lies in the first octant,
below the plane z  2 , with density  ( x, y, z )  x 2  z 2 .
(c) [Spherical coordinates] Solid between the hemispheres z  9  x 2  y 2 and
z  25  x 2  y 2 with x  0 and y  0 (which would be the parts of the hemispheres that lie
above the second quadrant of the xy plane. Density function  ( x, y, z ) 
1
x2  y 2  z 2
.