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SECTION 12.6 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES
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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
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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
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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.
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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.
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P6P612.6 CYLINDRICAL COORDINATES To convert from cylindrical to rectangular coordinates, we use: x = r cos y = r sin z = z
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P7P712.6 CYLINDRICAL COORDINATES To convert from rectangular to cylindrical coordinates, we use: r 2 = x 2 + y 2 tan = y/x z = z
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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).
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P9P912.6 Example 1(a) SOLUTION The point with cylindrical coordinates (2, 2 /3, 1) is plotted in Figure 3.
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P1012.6 Example 1(a) SOLUTION From Equations 1, its rectangular coordinates are: The point is ( – 1,, 1) in rectangular coordinates.
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P1112.6 Example 1(b) SOLUTION From Equations 2, we have:
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P1212.6 Example 1(b) SOLUTION Therefore, one set of cylindrical coordinates is Another is As with polar coordinates, there are infinitely many choices.
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P1312.6 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.
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P1412.6 CYLINDRICAL COORDINATES In cylindrical coordinates, this cylinder has the very simple equation r = c. This is the reason for the name “ cylindrical ” coordinates.
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P1512.6 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.
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P1612.6 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.
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P1712.6 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.
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P1812.6 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.
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P1912.6 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 ( )}
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P2012.6 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.
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P2112.6 Formula 4 This is the formula for triple integration in cylindrical coordinates.
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P2212.6 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 .
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P2312.6 TRIPLE INTEGN. IN CYL. COORDS. Figure 7 shows how to remember this.
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P2412.6 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.
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P2512.6 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
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P2612.6 Example 3 The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E.
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P2712.6 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}
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P2812.6 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.
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P2912.6 Example 3 SOLUTION So, from Formula 13 in Section 12.5, the mass of E is:
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P3012.6 Example 4 Evaluate
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P3112.6 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.
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P3212.6 Example 4 SOLUTION The lower surface of E is the cone Its upper surface is the plane z = 2.
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P3312.6 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.
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P3412.6 Example 4 SOLUTION
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