Single Integral
Domain: Area:
Double Integral
Domain: Area:
Volume: Mass:

Triple Integral

Domain:
Volume:
Mass:
 f ( x , y , z ) dV
E



Volume:

E
(1)
 
S
 
   
Mass: If density is
  x , y , z

, then mass =

E



Ex. Evaluate

0

2 
0
3 y  x
8
 x
2  y
2
2 
3 y
2 dzdxdy

Ex. Let E = solid bounded by the planes x = 0, x = 2, y = 0, z = 0, and y + z = 1. Find the volume of the solid.

Ex. Set up integrals to find the volume of the region formed by x = 4y 2 + 4z 2 and the plane x = 4.

Ex. Find the volume of the region bounded by y + z = 1, y = x 2 , and z = 0.

Average value of a function F over a region D =
1 volume of D

FdV
D
Ex. Find the average value of

2 , and z = 0.
 f ( x , y , z )
 x on the region

Do: 1. Set up an integral to find the volume of the region bounded by the coordinate planes and 2x + 3y + 6z = 12.
2. Set up an integral to find the volume of the region cut from the cylinder x 2 + y 2 = 4, the plane z = 0 and the plane x + z = 3.


Ex. Evaluate the integral by changing the order of integration in an appropriate way.
 1
0

3
1 z

0 ln 3
 e
2 x sin
 y
2 y
2 dxdydz



Ex. Sketch the solid whose volume is give by
1 1
 x 2

2 z 
0

0

0
.
Ex. Write 5 other iterated integrals that are equal to the given integral

0
1 
0 x 
0 y
(
,
,
)
.

Do. Write 6 different iterated integrals to find the volume of z

0, z
 y , x
2 
1
 y
.
the solid bounded by