SECTION 12.6 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES.

Slides:



Advertisements
Similar presentations
Polar Coordinates Objective: To look at a different way to plot points and create a graph.
Advertisements

16 MULTIPLE INTEGRALS.
MULTIPLE INTEGRALS MULTIPLE INTEGRALS 16.4 Double Integrals in Polar Coordinates In this section, we will learn: How to express double integrals.
MULTIPLE INTEGRALS Triple Integrals in Spherical Coordinates In this section, we will learn how to: Convert rectangular coordinates to spherical.
PARAMETRIC EQUATIONS AND POLAR COORDINATES 9. Usually, we use Cartesian coordinates, which are directed distances from two perpendicular axes. Here, we.
17 VECTOR CALCULUS.
16 MULTIPLE INTEGRALS.
16 MULTIPLE INTEGRALS.
MULTIPLE INTEGRALS Double Integrals over General Regions MULTIPLE INTEGRALS In this section, we will learn: How to use double integrals to.
17 VECTOR CALCULUS.
Chapter 15 – Multiple Integrals
Chapter 15 – Multiple Integrals
Chapter 12-Multiple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.
Multiple Integrals 12. Surface Area Surface Area In this section we apply double integrals to the problem of computing the area of a surface.
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
Cylindrical and Spherical Coordinates
Section 16.5 Integrals in Cylindrical and Spherical Coordinates.
Vectors and the Geometry of Space 9. 2 Announcement Wednesday September 24, Test Chapter 9 (mostly )
15.9 Triple Integrals in Spherical Coordinates
TRIPLE INTEGRALS IN SPHERICAL COORDINATES
Multiple Integration 14 Copyright © Cengage Learning. All rights reserved.
Section 16.4 Double Integrals In Polar Coordinates.
Vectors and the Geometry of Space 9. Functions and Surfaces 9.6.
Engineering Mechanics: Statics
Copyright © Cengage Learning. All rights reserved Double Integrals in Polar Coordinates.
Da Nang-09/2013 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Double Integrals in Polar Coordinates. We will learn about: How.
Chapter 15 – Multiple Integrals 15.4 Double Integrals in Polar Coordinates 1 Objectives:  Determine how to express double integrals in polar coordinates.
Multiple Integrals 12. Double Integrals over General Regions 12.3.
Copyright © Cengage Learning. All rights reserved Double Integrals over General Regions.
DOUBLE INTEGRALS OVER GENERAL REGIONS
ME 2304: 3D Geometry & Vector Calculus Dr. Faraz Junejo Double Integrals.
Sec 16.7 Triple Integrals in Cylindrical Coordinates In the cylindrical coordinate system, a point P is represented by the ordered triple (r, θ, z), where.
MAT 1236 Calculus III Section 15.8, 15.9 Triple Integrals in Cylindrical and Spherical Coordinates
Da Nang-03/2014 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Triple Integrals in Cylindrical Coordinates In this section, we.
Section 17.5 Parameterized Surfaces
Vector Functions 10. Parametric Surfaces Parametric Surfaces We have looked at surfaces that are graphs of functions of two variables. Here we.
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
DOUBLE INTEGRALS IN POLAR COORDINATES
APPLICATIONS OF DOUBLE INTEGRALS
Multiple Integration Copyright © Cengage Learning. All rights reserved.
Multiple Integrals 12.
Copyright © Cengage Learning. All rights reserved. 15 Multiple Integrals.
Triple Integrals in Cylindrical and Spherical Coordinates
SECTION 13.8 STOKES ’ THEOREM. P2P213.8 STOKES ’ VS. GREEN ’ S THEOREM  Stokes ’ Theorem can be regarded as a higher- dimensional version of Green ’
Vector Calculus CHAPTER 9.10~9.17. Ch9.10~9.17_2 Contents  9.10 Double Integrals 9.10 Double Integrals  9.11 Double Integrals in Polar Coordinates 9.11.
Multiple Integration 14 Copyright © Cengage Learning. All rights reserved.
Vectors and the Geometry of Space Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved. 16 Vector Calculus.
SECTION 12.5 TRIPLE INTEGRALS.
DOUBLE INTEGRALS OVER RECTANGLES
Double Integrals in Polar Coordinates. Sometimes equations and regions are expressed more simply in polar rather than rectangular coordinates. Recall:
Triple Integrals in Spherical and Cylindrical In rectangular coordinates: dV = dzdydx In cylindrical coordinates: dV = r dzdrdθ In spherical coordinates:
Copyright © Cengage Learning. All rights reserved.
Cylindrical and Spherical Coordinates
11 Vectors and the Geometry of Space
Copyright © Cengage Learning. All rights reserved.
Chapter 12 Math 181.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
14.7 Triple Integrals with Cylindrical and Spherical Coordinates
Copyright © Cengage Learning. All rights reserved.
Copyright © Cengage Learning. All rights reserved.
11 Vectors and the Geometry of Space
Copyright © Cengage Learning. All rights reserved.
Cylindrical and Spherical Coordinates
15.7 Triple Integrals.
Copyright © Cengage Learning. All rights reserved.
Presentation transcript:

SECTION 12.6 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES

P2P212.6 POLAR COORDINATES  In plane geometry, the polar coordinate system is used to give a convenient description of certain curves and regions. See Section 9.3

P3P312.6 POLAR COORDINATES  Figure 1 enables us to recall the connection between polar and Cartesian coordinates. If the point P has Cartesian coordinates (x, y) and polar coordinates (r,  ), then x = r cos  y = r sin  r 2 = x 2 + y 2 tan  = y/x

P4P412.6 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.  As we will see, some triple integrals are much easier to evaluate in cylindrical coordinates.

P5P512.6 CYLINDRICAL COORDINATES  In the cylindrical coordinate system, a point P in three-dimensional (3-D) space is represented by the ordered triple (r, , z), where: r and  are polar coordinates of the projection of P onto the xy – plane. z is the directed distance from the xy-plane to P. See Figure 2.

P6P612.6 CYLINDRICAL COORDINATES  To convert from cylindrical to rectangular coordinates, we use: x = r cos  y = r sin  z = z

P7P712.6 CYLINDRICAL COORDINATES  To convert from rectangular to cylindrical coordinates, we use: r 2 = x 2 + y 2 tan  = y/x z = z

P8P812.6 Example 1 a)Plot the point with cylindrical coordinates (2, 2  /3, 1) and find its rectangular coordinates. b)Find cylindrical coordinates of the point with rectangular coordinates (3, – 3, – 7).

P9P912.6 Example 1(a) SOLUTION  The point with cylindrical coordinates (2, 2  /3, 1) is plotted in Figure 3.

P Example 1(a) SOLUTION  From Equations 1, its rectangular coordinates are: The point is ( – 1,, 1) in rectangular coordinates.

P Example 1(b) SOLUTION  From Equations 2, we have:

P Example 1(b) SOLUTION  Therefore, one set of cylindrical coordinates is  Another is As with polar coordinates, there are infinitely many choices.

P 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 x 2 + y 2 = c 2 is the z-axis.

P CYLINDRICAL COORDINATES In cylindrical coordinates, this cylinder has the very simple equation r = c. This is the reason for the name “ cylindrical ” coordinates.

P Example 2  Describe the surface whose equation in cylindrical coordinates is z = r. SOLUTION 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  doesn ’ t appear, it can vary.

P Example 2 SOLUTION  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.

P Example 2 SOLUTION  From the first equation in Equations 2, we have: z 2 = r 2 = x 2 + y 2  We recognize the equation z 2 = x 2 + y 2 (by comparison with Table 1 in Section 10.6) as being a circular cone whose axis is the z-axis.  See Figure 5.

P EVALUATING TRIPLE INTEGRALS WITH CYLINDRICAL COORDINATES  Suppose that E is a type 1 region whose projection D on the xy- plane is conveniently described in polar coordinates.  See Figure 6.

P EVALUATING TRIPLE INTEGRALS  In particular, suppose that f is continuous and E = {(x, y, z) | (x, y)  D, u 1 (x, y) ≤ z ≤ u 2 (x, y)} where D is given in polar coordinates by: D = {(r,  ) |  ≤  ≤ , h 1 (  ) ≤ r ≤ h 2 (  )}

P EVALUATING TRIPLE INTEGRALS  We know from Equation 6 in Section 12.5 that:  However, we also know how to evaluate double integrals in polar coordinates.  In fact, combining Equation 3 with Equation 3 in Section 12.3, we obtain the following formula.

P Formula 4  This is the formula for triple integration in cylindrical coordinates.

P TRIPLE INTEGN. IN CYL. COORDS.  It says that we convert a triple integral from rectangular to cylindrical coordinates by: Writing x = r cos , y = r sin . Leaving z as it is. Using the appropriate limits of integration for z, r, and . Replacing dV by r dz dr d .

P TRIPLE INTEGN. IN CYL. COORDS.  Figure 7 shows how to remember this.

P 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 x 2 + y 2.

P Example 3  A solid lies within: The cylinder x 2 + y 2 = 1 Below the plane z = 4 Above the paraboloid z = 1 – x 2 – y 2

P Example 3  The density at any point is proportional to its distance from the axis of the cylinder.  Find the mass of E.

P Example 3 SOLUTION  In cylindrical coordinates the cylinder is r = 1 and the paraboloid is z = 1 – r 2.  So, we can write: E ={(r, , z)| 0 ≤  ≤ 2 , 0 ≤ r ≤ 1, 1 – r 2 ≤ z ≤ 4}

P Example 3 SOLUTION  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.

P Example 3 SOLUTION  So, from Formula 13 in Section 12.5, the mass of E is:

P Example 4  Evaluate

P Example 4 SOLUTION  This iterated integral is a triple integral over the solid region  The projection of E onto the xy-plane is the disk x 2 + y 2 ≤ 4.

P Example 4 SOLUTION  The lower surface of E is the cone  Its upper surface is the plane z = 2.

P Example 4 SOLUTION  That region has a much simpler description in cylindrical coordinates: E = {(r, , z) | 0 ≤  ≤ 2 , 0 ≤ r ≤ 2, r ≤ z ≤ 2} Thus, we have the following result.

P Example 4 SOLUTION