SECTION 12.6

1. SECTION 12.6 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES

2. 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 12.6

3. 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, ), thenx = r cos  y = r sinr2 = x2 + y2 tan  = y/x 12.6

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. • As we will see, some triple integrals are much easier to evaluate in cylindrical coordinates. 12.6

5. 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. 12.6

6. CYLINDRICAL COORDINATES • To convert from cylindrical to rectangular coordinates, we use:x = r cos y = r sin z = z 12.6

7. CYLINDRICAL COORDINATES • To convert from rectangular to cylindrical coordinates, we use:r2 = x2 + y2 tan  = y/xz = z 12.6

8. Example 1 • Plot the point with cylindrical coordinates (2, 2p/3, 1) and find its rectangular coordinates. • Find cylindrical coordinates of the point with rectangular coordinates (3, –3, –7). 12.6

9. Example 1(a) SOLUTION • The point with cylindrical coordinates (2, 2p/3, 1) is plotted in Figure 3. 12.6

10. Example 1(a) SOLUTION • From Equations 1, its rectangular coordinates are: • The point is (–1, , 1) in rectangular coordinates. 12.6

11. Example 1(b) SOLUTION • From Equations 2, we have: 12.6

12. Example 1(b) SOLUTION • Therefore, one set of cylindrical coordinates is • Another is • As with polar coordinates, there are infinitely many choices. 12.6

13. 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. 12.6

14. CYLINDRICAL COORDINATES • In cylindrical coordinates, this cylinder has the very simple equation r = c. • This is the reason for the name “cylindrical” coordinates. 12.6

15. 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. 12.6

16. 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. 12.6

17. Example 2 SOLUTION • From the first equation in Equations 2, we have: z2 = r2 = x2 + y2 • We recognize the equation z2 = x2 + y2(by comparison with the table in Section 10.6) as being a circular cone whose axis is the z-axis. • See Figure 5. 12.6

18. 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. 12.6

19. 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, ) | a ≤  ≤ b, h1() ≤ r ≤ h2()} 12.6

20. 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. 12.6

21. Formula 4 • This is the formula for triple integration in cylindrical coordinates. 12.6

22. 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. 12.6

23. TRIPLE INTEGN. IN CYL. COORDS. • Figure 7 shows how to remember this. 12.6

24. 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. 12.6

25. Example 3 • A solid lies within: • The cylinder x2 + y2 = 1 • Below the plane z = 4 • Above the paraboloid z = 1 –x2–y2 12.6

26. Example 3 • The density at any point is proportional to its distance from the axis of the cylinder. • Find the mass of E. 12.6

27. Example 3 SOLUTION • In cylindrical coordinates, the cylinder is r = 1 and the paraboloid is z = 1 –r2. • So, we can write:E = {(r, , z)| 0 ≤≤ 2p, 0 ≤r≤ 1, 1 –r2≤z≤ 4} 12.6

28. 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. 12.6

29. Example 3 SOLUTION • So, from Formula 13 in Section 12.5, the mass of E is: 12.6

30. Example 4 • Evaluate 12.6

31. 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 x2 + y2≤ 4. 12.6

32. Example 4 SOLUTION • The lower surface of E is the cone • Its upper surface is the plane z = 2. 12.6

33. Example 4 SOLUTION • That region has a much simpler description in cylindrical coordinates:E = {(r, , z) | 0 ≤ ≤ 2p, 0 ≤ r≤ 2, r ≤ z≤ 2} • Thus, we have the following result. 12.6

34. Example 4 SOLUTION 12.6