TRIPLE INTEGRALS
Single Integrals featured infinitesimal line
segments; double integrals featured infinitesimal
rectangles. Not surprisingly, therefore, triple
integrals feature infinitesimal parallelopipeds
wherein a given volume or finite solid, " G ", is
decomposed into those tiny " boxes " and then
multiplied or " weighted " by the function of 3
variables resident in the integrand.
Example # 1: Evaluate the iterated integral.
3
x
⌠ ⌠
⎮ ⎮
⌡1 ⌡x
2
ln ( z)
⌠
⎮
⌡0
y
x ⋅ e dy dz dx
Multiple integrals are evaluated starting with the
integration variable whose differential is
innermost. In this caes, that integration variable
is " y ". So, we first compute this single integral.
ln ( z)
⌠
⎮
⌡0
y
(
x ⋅ e dy = x e
ln ( z)
−e
0
) = x⋅ (z − 1)
Page 1 of 13
Now compute the integral with respect to the
middle integration variable, namely " z ".
x
⌠
⎮
⌡x
2
x
2
x
2
⌠
⎮
⌡x
⌠
⎮
⌡x
x
⌠
x ⋅ ( z − 1) dz = x ⋅ ⎮
⌡x
2
x
⌠
z dz − x ⋅ ⎮
⌡x
(
)
2
1 dz
(
)
1
4
2
2
x ⋅ ( z − 1) dz = x ⋅ ⎛⎜ ⎟⎞ ⋅ x − x − x ⋅ x − x
⎝ 2⎠
5
3 3
x
2
x ⋅ ( z − 1) dz =
− ⋅x + x
2
2
Finally, compute the integral with restect to " x ".
3
⌠ ⎛ 5
3 3
118
⎮ x
2⎞
⎜
⎟
dx =
−
⋅
x
+
x
⎮ ⎝ 2
2
3
⎠
⌡1
3
x
⌠ ⌠
⎮ ⎮
⌡1 ⌡x
2
ln ( z)
⌠
⎮
⌡0
y
x ⋅ e dy dz dx =
118
3
Page 2 of 13
Example # 2: Evaluate the triple integral over "G",
where "G" is the solid enclosed by the plane:
z = y, the xy-plane, and the parabolic cylinder:
2
y=1−x .
⌠
⎮ y dv
⌡
"G" Region
z
y
1
⌠
⎮
⌡− 1
x
1−x
⌠
⎮
⌡0
2
y
⌠
⎮ y dz dy dx
⌡0
Page 3 of 13
y
⌠
2
⎮ y dz = y
⌡0
1−x
⌠
⎮
⌡0
⌠
⎮
⎮
⎮
⌡
1
2
y
⌠
⎮
⌡− 1
(
1−x )
dy =
3
1−x
⌠
⎮
⌡0
3
3
( 1 − x 2 ) 3 dx =
−1
1
2
2
2
32
105
y
⌠
32
⎮ y dz dy dx =
⌡0
105
Example # 3: Use a Triple Integral to find the
volume of the solid wedge in the first quadrant
2
2
that is cut from the solid cylinder: y + z ≤ 1 by
the planes: y = x and x = 0.
Page 4 of 13
Wedge
z
x
y
2
1
1−y
y
⌠ ⌠
⌠
⎮ ⎮
⌡0 ⌡0
⎮ 1 dx dz dy
⌡0
y
⌠
⎮ 1 dx = y
⌡0
Page 5 of 13
⌠ 1−y
⎮
⌡0
2
y dz = y ⋅ 1 − y
2
1
⌠
1
⎮ y ⋅ 1 − y 2 dy =
⌡0
3
2
1
1−y
y
⌠ ⌠
⌠
⎮ ⎮
⌡0 ⌡0
1
⎮ 1 dx dz dy =
3
⌡0
Example # 4: Set-up a Triple Integral for the
volume of the solid enclosed bewteen the
2
2
2
2
surfaces: z = 3 ⋅ x + y and z = 8 − x − y .
Page 6 of 13
Volume between Surfaces
z
x
y
2
2
2
2
2
3⋅ x + y = 8 − x − y
2
2⋅ x + y = 4
y = − 4 − 2⋅ x
⌠
V=⎮
⌡
−
2
⌠ 4−2 ⋅ x
2
2
⎮
⌡
−
4 − 2⋅ x
y=
2
4−2 ⋅ x
2
⌠
⎮
⌡
2
8−x −y
2
2
1 dz dy dx
2
3 ⋅ x +y
2
Page 7 of 13
Example # 5(a): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
2
2
3
9−z ⌠ 9−y −z
⌠ ⌠
⎮ ⎮
⌡0 ⌡0
2
f ( x , y , z ) dx dy dz
⎮
⌡0
Enclosed Volume
z
x
y
2
2
3
9−x ⌠ 9−x −y
⌠ ⌠
⎮ ⎮
⌡0 ⌡0
⎮
⌡0
2
f ( x , y , z ) dz dy dx
Page 8 of 13
Example # 5(b): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
x
⌠ ⌠ ⌠2
4
2
⎮ ⎮ ⎮ f ( x , y , z ) dy dz dx
⌡0 ⌡0 ⌡0
Enclosed Volume
z
x
y
x
⌠ ⌠2
4
2
⎮ ⎮ ⌠
⎮ f ( x , y , z ) dz dy dx
⌡0 ⌡0 ⌡0
Page 9 of 13
Example # 5(c): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
4−y
4
⌠ ⌠
⎮ ⎮
⌡0 ⌡0
⌠ z
⎮ f ( x , y , z ) dx dz dy
⌡0
Enclosed Volume
z
x
y
2
4−x
⌠ ⌠
⎮ ⎮
⌡0 ⌡0
2
⌠
⎮
⌡
4−y
x
f ( x , y , z ) dz dy dx
2
Page 10 of 13
Example # 5(d): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
"G" Region
z
y
x
2
4−y
⌠ ⌠ ⌠
5
2
⎮ ⎮ ⎮
⌡0 ⌡0 ⌡0
1 dx dy dz
2
2
4−x
5
⌠ ⌠
⌠
⎮ ⎮
⌡0 ⌡0
⎮ 1 dz dy dx
⌡0
Page 11 of 13
Example # 5(e): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
"G" Region
z
y
x
9 3− x z
⌠ ⌠
⌠
⎮ ⎮
⎮ 1 dy dz dx
⌡
⌡0 0
⌡0
9 3− x 3− x
⌠
⌠ ⌠
⎮
⎮
⎮
1 dz dy dx
⌡y
⌡0 ⌡0
Page 12 of 13
Example # 5(f): Express the given integral as an
equivalent integral in which the z-integration is
performed first, the y-integration second, and the
x-integration last.
"G" Region
z
y
x
4
8−y
2
4−x
⌠ ⌠
⎮ ⎮
⌡0 ⌡y
⌠ ⌠
⎮ ⎮
⌡0 ⌡0
⌠ 4−y
⎮
1 dx dz dy
⌡0
2
8−y
⌠
⎮
⌡y
1 dz dy dx
Page 13 of 13