Let f (x, y) be the mass density at (x, y) in kg/m2
Suppose we wish to find the total mass in a two dimensional
region R. Start by dividing R into a grid. The mass in one
section is:
f (x, y) dA
The total mass from all sections is:
∫∫
∫∫
f (x, y) dA =
f (x, y) dx dy
R
R
We may generalize to three dimensions.
Let f (x, y, z) be the density at (x, y, z) in kg/m3
Suppose we wish to find the total mass in a three dimensional
region B.
The mass of one section is (approximately)
f (x, y, z) dx dy dz
The total mass in region B is the limit of the sum as the number
of sections goes to infinity.
∫∫∫
Mass(B) =
f (x, y, z) dx dy dz
B
Centroid
1
x=
Area(R)
∫∫
x dA
R
1
y=
Area(R)
∫∫
y dA
R
Centroid of a 3 dimensional region T
∫∫∫
∫∫∫
1
1
x=
x dV
y=
y dV
Vol(T )
Vol(T
)
T
T
∫∫∫
1
z=
z dV
Vol(T )
T
∫
b
1 dx = b − a
a
∫∫
Area(R) =
1 dA
R
∫∫∫
Vol(T ) =
1 dV
T
Find the volume under the portion of the plane
z = 1− y4 that is directly over the triangle with vertices (0, 0, 0),
(2, 4, 0) and (0, 4, 0)
Find the volume bounded by the planes:
y = 0,
z=0
x = 2y + 2z
x+y+z =3
Find the volume of the region bounded from below
by z = x2 + y 2 and from above by z = 1.
∫
√
2
√
− 2
∫
√
2−x2
∫
2−x2 −y 2
1
dz dy dx
√
2 + y2
1
+
x
2
− 2−x 0
∫ ?∫ ?∫ ?
1
=
dx dy dz
2 + y2
1
+
x
?
?
?
Reverse the order of integration:
∫
√
2
∫
√
2−x2
∫
√
√
− 2 − 2−x2 0
=
2−x2 −y 2
∫
1
dz dy dx
1 + x2 + y 2
∫
∫
1
dx dy dz
1 + x2 + y 2
Express the volume of the region bounded by z = ∫∫∫
x2 + y 2 and
z = 4 − 3(x2 + y 2 ) as a triple integral in the form
1 dV
The volume under the cone z = 1 −
centroid.
√
x2 + y 2 is
π
3.
Find the
Find the volume of the region inside the sphere x2 +y 2 +z 2 = 4
and above the plane z = 1.
Find the volume of the region bounded by:
z + x2 = 9
y+z =4
y=0
y=4
Let T be the tetrahedron with the vertices
∫∫∫(0, 0, 0), (0, 0, 1),
(1, −1, 0) and (1, 1, 0). Calculate Vol(T ) =
1 dz dy dx
T
Ω is the section inside
∫∫∫the cone z =
Calculate Vol(Ω) =
1 dV
Ω
√
x2 + y 2 for 1 ≤ z ≤ 2.
2
2
2
Let Q be the
region
inside
x
+
y
+
z
= 1 for x, y, z ≥ 0.
∫∫∫
Calculate
z dV
Q