Vector Calculus for Engineers
CME100, Fall 2004
Handout #3
Double and Triple Integrals in Cartesian, Cylindrical,
and Spherical Coordinates with Applications
1. Calculate the volume under the plane z  4  x  y
over the rectangular region R: 0  x  2, 0  y  1
2. Find the volume of the prism whose base is the triangle
in the xy-plane bounded by the x-axis and the lines
y  x and x  1, and whose top lies in the plane
z  3 x  y
sin x
dA over the triangle in the xy-plane
x
R
bounded by the x-axis, y  x , and x  1
3. Calculate

4. A thin plate covers the triangular region bounded by the
x-axis and the lines x  1 and y  2 x in the first
quadrant. The density is given by  ( x, y )  6 x  6 y  6 .
Find the mass and the center of mass of the plate.
5. For the region in the previous example find the moment
of inertia I x
6. Find the center of mass of a semi-circular disk bounded
by the x-axis and y  R 2  x 2
a) using Cartesian coordinates
b) using polar coordinates
7. Using triple integrals, find the volume of the region
enclosed by the plane x  y  z  1 and coordinate
planes x  0 , y  0 , and z  0
8. Find the center of mass of a solid of constant density 
bounded by the disk x 2  y 2  4 in the plane z  0 and
above by the paraboloid z  4  x 2  y 2
9. Find moments of inertia I x , I y and I z for the
rectangular solid of constant density  and dimensions
in the x, y and z directions of a, b and c respectively.
10. Using cylindrical coordinates find the volume and the
center of mass of a cone with uniform density formed
by z 2  x 2  y 2 and z  1
11. Using cylindrical coordinates find the volume above the
xy-plane bounded by x 2  y 2  1 and x  y  z  2
12. Using spherical coordinates find the center of mass of a
hemisphere of radius R
13. Find the moment of inertia of a uniform solid sphere of
mass M and radius R
14. Using transformation of coordinates find the area
enclosed by y  x , y  2 x , xy  1 and xy  2
15. Find the Jacobian of transformation between:
a) Cartesian and polar coordinates
b) Cartesian and cylindrical coordinates
c) Cartesian and spherical coordinates