Lesson 22: Triple Integrals
21
. . . . . . Section 12.7 Triple Integrals Math 21a April 4, 2008 Announcements ◮ Office hours Tuesday, Wednesday 2–4pm SC 323 ◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b . . Image: Flickr user The K Team
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You knew this was coming. From double integrals over plane regions we move onward to triple integrals over solid regions. The visualization is a little harder, but the calculus not that much.
Transcript of Lesson 22: Triple Integrals
. . . . . .
Section 12.7Triple Integrals
Math 21a
April 4, 2008
Announcements
◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b
..Image: Flickr user The K Team
. . . . . .
Announcements
◮ Office hours Tuesday, Wednesday 2–4pm SC 323◮ Problem Sessions: Mon, 8:30; Thur, 7:30; SC 103b
. . . . . .
Outline
Last time: Surface Area
Triple integrals over boxesThe Riemann sumFubini’s Theorem for triple integrals
Triple integrals over solid regions
Worksheet
Next Time
. . . . . .
Last Time: Surface Area
◮ if r : D → S is a parametrization, the surface area of S is
A(S) =
∫∫D
|ru × rv| dA
◮ If S is the graph of f(x, y) over D, then
A(S) =
∫∫D
√1 +
(∂f∂x
)2
+
(∂f∂y
)2
dA
◮ If S is the graph of y = f(x) over [a, b] rotated about the x-axis,then
A(S) = 2π
∫ b
af(x)
√1 + f′(x)2 dx
. . . . . .
Outline
Last time: Surface Area
Triple integrals over boxesThe Riemann sumFubini’s Theorem for triple integrals
Triple integrals over solid regions
Worksheet
Next Time
. . . . . .
The Riemann sum
To integrate a function f(x, y, z) over the three-dimensional box
B = [a, b] × [c, d] × [r, s]
= { (x, y, z) | a ≤ x ≤ b, c ≤ y ≤ d, r ≤ z ≤ s }
◮ Divide up [a, b] into ℓ pieces, [c, d] into m pieces, [r, s] into npieces
◮ choose a sample point (x∗ijk, y∗ijk, z∗ijk) in each sub-box◮ form the Riemann sum
Sℓmn =ℓ∑
i=1
m∑j=1
n∑k=1
f(x∗ijk, y∗ijk, z∗ijk)∆x ∆y ∆z
◮ take the limit!
. . . . . .
Definition
∫∫∫B
f(x, y, z) dV = limℓ,m,n→∞
ℓ∑i=1
m∑j=1
n∑k=1
f(x∗ijk, y∗ijk, z∗ijk)∆x ∆y ∆z
. . . . . .
TheoremIf f is continuous on the rectangular box
B = [a, b] × [c, d] × [r, s]
then ∫∫∫B
f(x, y, z) dV =
∫ s
r
∫ d
c
∫ b
af(x, y, z) dx dy dz
. . . . . .
ExampleEvaluate the triple integral∫∫∫
B
2xey sin z dV, where B = [1, 2] × [0, 1] × [0, π]
Answer
∫∫∫B
2xey sin z dV =
∫ π
0
∫ 1
0
∫ 2
12xey sin z dx dy dz = 6(e − 1)
. . . . . .
ExampleEvaluate the triple integral∫∫∫
B
2xey sin z dV, where B = [1, 2] × [0, 1] × [0, π]
Answer
∫∫∫B
2xey sin z dV =
∫ π
0
∫ 1
0
∫ 2
12xey sin z dx dy dz = 6(e − 1)
. . . . . .
Outline
Last time: Surface Area
Triple integrals over boxesThe Riemann sumFubini’s Theorem for triple integrals
Triple integrals over solid regions
Worksheet
Next Time
. . . . . .
Motivation
◮ We can define triple integrals over non-box regions◮ For double integrals, we found some regions which converted
easily to iterated integrals (type I and type II)◮ What’s the three-dimensional analogue of such simple regions
. . . . . .
DefinitionA solid region E is said to be of type 1 if it lies between the graphsof two continuous function of x and y:
E = { (x, y, z) | (x, y) ∈ D, u1(x, y) ≤ z ≤ u2(x, y) }
FactLet E be as above, and f a continuous function. Then∫∫∫
E
f(x, y, z) dV =
∫∫D
[∫ u2(x,y)
u1(x,y)f(x, y, z) dz
]dA
. . . . . .
DefinitionA solid region E is said to be of type 1 if it lies between the graphsof two continuous function of x and y:
E = { (x, y, z) | (x, y) ∈ D, u1(x, y) ≤ z ≤ u2(x, y) }
FactLet E be as above, and f a continuous function. Then∫∫∫
E
f(x, y, z) dV =
∫∫D
[∫ u2(x,y)
u1(x,y)f(x, y, z) dz
]dA
. . . . . .
If D is of type I, we can further reduce the integral. Suppose
D = (x, y)a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)
Then∫∫∫E
f(x, y, z) dV =
∫∫D
[∫ u2(x,y)
u1(x,y)f(x, y, z) dz
]dA
=
∫ b
a
∫ g2(x)
g1(x)
∫ u2(x,y)
u1(x,y)f(x, y, z) dz dy dx
. . . . . .
A type 2 region is of the form
E = { (x, y, z) | (y, z) ∈ D, u1(y, z) ≤ x ≤ u2(y, z) }
The integral of f over E can be computed as∫∫∫E
f(x, y, z) dV =
∫∫D
∫ u2(y,z)
u1(y,z)f(x, y, z) dx dA
. . . . . .
A type 3 region is of the form
E = { (x, y, z) | (x, z) ∈ D, u1(x, z) ≤ y ≤ u2(x, z) }
The integral of f over E can be computed as∫∫∫E
f(x, y, z) dV =
∫∫D
∫ u2(x,z)
u1(x,z)f(x, y, z) dy dA
. . . . . .
Outline
Last time: Surface Area
Triple integrals over boxesThe Riemann sumFubini’s Theorem for triple integrals
Triple integrals over solid regions
Worksheet
Next Time
. . . . . .
Worksheet
.
.Image: Erick Cifuentes
. . . . . .
Outline
Last time: Surface Area
Triple integrals over boxesThe Riemann sumFubini’s Theorem for triple integrals
Triple integrals over solid regions
Worksheet
Next Time
. . . . . .
Next time:More Triple Integrals
