multiple integrals 16. 2 polar coordinates in plane geometry, the polar coordinate system is used to...
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MULTIPLE INTEGRALSMULTIPLE INTEGRALS
16
2
POLAR COORDINATES
In plane geometry, the polar coordinate
system is used to give a convenient
description of certain curves and regions.
See Section 11.3
3
POLAR COORDINATES
The figure enables us
to recall the connection
between polar and
Cartesian coordinates.
If the point P has Cartesian coordinates (x, y) and polar coordinates (r, θ), then
x = r cos θ y = r sin θ
r2 = x2 + y2 tan θ = y/x
Fig. 16.7.1, p. 1036
4
CYLINDRICAL COORDINATES
In three dimensions there is a coordinate
system, called cylindrical coordinates,
that:
Is similar to polar coordinates.
Gives a convenient description of commonly occurring surfaces and solids.
5
16.7Triple Integrals in
Cylindrical Coordinates
MULTIPLE INTEGRALS
In this section, we will learn about:
Cylindrical coordinates and
using them to solve triple integrals.
6
CYLINDRICAL COORDINATES
In the cylindrical coordinate system, a point P
in three-dimensional (3-D) space is
represented by the ordered triple (r, θ, z),
where: r and θ are polar
coordinates of the projection of P onto the xy–plane.
z is the directed distance from the xy-plane to P.
Fig. 16.7.2, p. 1037
7
CYLINDRICAL COORDINATES
To convert from cylindrical to rectangular
coordinates, we use:
x = r cos θ
y = r sin θ
z = z
Equations 1
8
CYLINDRICAL COORDINATES
To convert from rectangular to cylindrical
coordinates, we use:
r2 = x2 + y2
tan θ = y/x
z = z
Equations 2
9
CYLINDRICAL COORDINATES
a. Plot the point with cylindrical
coordinates (2, 2π/3, 1) and find its
rectangular coordinates.
b. Find cylindrical coordinates of the point
with rectangular coordinates (3, –3, –7).
Example 1
10
CYLINDRICAL COORDINATES
The point with cylindrical coordinates
(2, 2π/3, 1) is plotted here.
Example 1 a
Fig. 16.7.3, p. 1037
11
CYLINDRICAL COORDINATES
From Equations 1, its rectangular coordinates
are:
The point is (–1, , 1) in rectangular coordinates.
2 12cos 2 1
3 2
2 32sin 2 3
3 2
1
x
y
z
3
Example 1 a
12
CYLINDRICAL COORDINATES
From Equations 2, we have:
Example 1 b
2 23 ( 3) 3 2
3 7tan 1, so 2
3 47
r
n
z
13
CYLINDRICAL COORDINATES
Therefore, one set of cylindrical coordinates
is:
Another is:
As with polar coordinates, there are infinitely many choices.
(3 2,7 / 4, 7)
(3 2, / 4, 7)
Example 1 b
14
CYLINDRICAL COORDINATES
Cylindrical coordinates are useful in problems
that involve symmetry about an axis, and
the z-axis is chosen to coincide with this axis
of symmetry.
For instance, the axis of the circular cylinder with Cartesian equation x2 + y2 = c2 is the z-axis.
15
CYLINDRICAL COORDINATES
In cylindrical coordinates, this cylinder has the very simple equation r = c.
This is the reason for the name “cylindrical” coordinates.
Fig. 16.7.4, p. 1037
16
CYLINDRICAL COORDINATES
Describe the surface whose equation
in cylindrical coordinates is z = r.
The equation says that the z-value, or height, of each point on the surface is the same as r, the distance from the point to the z-axis.
Since θ doesn’t appear, it can vary.
Example 2
17
CYLINDRICAL COORDINATES
So, any horizontal trace in the plane z = k
(k > 0) is a circle of radius k.
These traces suggest the surface is a cone.
This prediction can be confirmed by converting the equation into rectangular coordinates.
Example 2
18
CYLINDRICAL COORDINATES
From the first equation in Equations 2,
we have:
z2 = r2 = x2 + y2
Example 2
19
CYLINDRICAL COORDINATES
We recognize the equation z2 = x2 + y2
(by comparison with the table in Section 13.6)
as being a circular cone whose axis is
the z-axis.
Example 2
Fig. 16.7.5, p. 1038
20
EVALUATING TRIPLE INTEGS. WITH CYL. COORDS.
Suppose that E is a type 1 region whose
projection D on the xy-plane is conveniently
described in
polar coordinates.
Fig. 16.7.6, p. 1038
21
EVALUATING TRIPLE INTEGRALS
In particular, suppose that f is continuous
and
E = {(x, y, z) | (x, y) D, u1(x, y) ≤ z ≤ u2(x, y)}
where D is given in polar coordinates by:
D = {(r, θ) | α ≤ θ ≤ β, h1(θ) ≤ r ≤ h2(θ)}
22
EVALUATING TRIPLE INTEGRALS
We know from Equation 6 in Section 16.6
that:
2
1
( , )
( , )
( , , )
, ,
E
u x y
u x yD
f x y z dV
f x y z dz dA
Equation 3
23
EVALUATING TRIPLE INTEGRALS
However, we also know how to evaluate
double integrals in polar coordinates.
In fact, combining Equation 3 with Equation 3
in Section 16.4, we obtain the following
formula.
24
TRIPLE INTEGN. IN CYL. COORDS.
f x, y, z dVE
f r cos ,r sin , z r dz dr du1 r cos ,r sin
u2 r cos ,r sin h1 ( )
h2 ( )
Formula 4
This is the formula for triple integration
in cylindrical coordinates.
25
TRIPLE INTEGN. IN CYL. COORDS.
It says that we convert a triple integral from
rectangular to cylindrical coordinates by:
Writing x = r cos θ, y = r sin θ.
Leaving z as it is.
Using the appropriate limits of integration for z, r, and θ.
Replacing dV by r dz dr dθ.
26
TRIPLE INTEGN. IN CYL. COORDS.
The figure shows how to
remember this.
Fig. 16.7.7, p. 1038
27
TRIPLE INTEGN. IN CYL. COORDS.
It is worthwhile to use this formula:
When E is a solid region easily described in cylindrical coordinates.
Especially when the function f(x, y, z) involves the expression x2 + y2.
28
EVALUATING TRIPLE INTEGRALS
A solid lies within:
The cylinder x2 + y2 = 1
Below the plane z = 4
Above the paraboloid z = 1 – x2 – y2
Example 3
Fig. 16.7.8, p. 1039
29
EVALUATING TRIPLE INTEGRALS
The density at any point is proportional to
its distance from the axis of the cylinder.
Find the mass of E.
Example 3
Fig. 16.7.8, p. 1039
30
EVALUATING TRIPLE INTEGRALS
In cylindrical coordinates, the cylinder is r = 1
and the paraboloid is z = 1 – r2.
So, we can write:
E =
{(r, θ, z)| 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 1, 1 – r2 ≤ z ≤ 4}
Example 3
31
EVALUATING TRIPLE INTEGRALS
As the density at (x, y, z) is proportional
to the distance from the z-axis, the density
function is:
where K is the proportionality constant.
2 2, ,f x y z K x y Kr
Example 3
32
EVALUATING TRIPLE INTEGRALS
So, from Formula 13 in Section 16.6,
the mass of E is:
2
2 2
2 1 4
0 0 1
2 1 2 2
0 0
2 1 2 4
0 0
153
0
( )
4 1
3
122
5 5
E
r
m K x y dV
Kr r dz dr d
Kr r dr d
K d r r dr
r KK r
Example 3
33
EVALUATING TRIPLE INTEGRALS
Evaluate
Example 4
2
2 2 2
2 4 2 2 2
2 4
x
x x yx y dz dy dx
34
EVALUATING TRIPLE INTEGRALS
This iterated integral is a triple integral
over the solid region
The projection of E onto the xy-plane
is the disk x2 + y2 ≤ 4.
2 2 2 2{ , , | 2 2, 4 4 , 2}
E
x y z x x y x x y z
Example 4
35
EVALUATING TRIPLE INTEGRALS
The lower surface of E is the cone
Its upper surface is
the plane z = 2.
Example 4
2 2z x y
Fig. 16.7.9, p. 1039
36
EVALUATING TRIPLE INTEGRALS
That region has a much simpler description
in cylindrical coordinates:
E =
{(r, θ, z) | 0 ≤ θ ≤ 2π, 0 ≤ r ≤ 2, r ≤ z ≤ 2}
Thus, we have the following result.
Example 4
37
EVALUATING TRIPLE INTEGRALS
2
2 2 2
2 4 2 2 2
2 4
2 2 22 2 2
0 0
2 2 3
0 0
24 51 12 5 0
165
2
2
x
x x y
rE
x y dz dy dx
x y dV r r dz dr d
d r r dr
r r
Example 4