Lecture 13-7 - Defn Spherical Coordinates A point P in...

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13.7 Triple Integrals in Cylindrical and Spher- ical Coordinates Using cylindrical or spherical coordinates can be helpful for integral calcula- tions over regions involving cylinders, cones and spheres. Cylindrical Coordinates (i.e. Polar Coordinates in 3D) Defn. Cylindrical Coordinates A point P in cylindrical coordinates is ( r, θ, z ) where 1. 2. 1
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Conversions (Rectangular Cylindrical) Graphing Examples 1. r = a 2. θ = π 4 2
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Cylindrical Triple Integrals Defn. Cylindrical Triple Integral A cylindrical triple integral on a surface D is where D is given in cylindrical coordinates. Tips 1. 2. 3
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Example 1 Using cylindrical coordinates, find the volume of the solid bounded by x 2 + y 2 = z 2 and x 2 + y 2 = 4 4
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Example 2 Set up ZZZ E xdV in cylindrical coordinates where E is the solid bounded by x 2 + y 2 = z and z = 18 - x 2 - y 2 . Example 3 Convert the following integral to cylindrical coordinates Z 1 - 1 Z 1 - x 2 - 1 - x 2 Z 1 x 2 + y 2 dzdydx. 5
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Do. Set up an integral in cylindrical coordinates to find the volume of the region inside the sphere x 2 + y 2 + z 2 = 2 and outside the cylinder x 2 + y 2 = 1. 6
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Spherical Coordinates
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Unformatted text preview: Defn. Spherical Coordinates A point P in spherical coordinates is ( ρ, φ, θ ) where 1. 2. 3. 7 Equations Relating Spherical, Cylindrical and Rectangular Coor-dinates Example Convert z = p x 2 + y 2 to spherical coordinates. Graphing Examples 1. ρ = 3 2. θ = 3 π 4 3. φ = 3 π 4 4. φ = π 8 Spherical Triple Integrals Defn. Spherical Triple Integral A spherical triple integral on a surface D is where D is given in spherical coordinates. 9 Tips 1. 2. 3. 10 Example 1 Find the volume bounded by x 2 + y 2 = z 2 and x 2 + y 2 = 4. 11 Example 2 Evaluate ZZZ E y dV where E is the hollow sphere whose outside radius is √ 2 and inside radius is 1. 12 Do. Find the volume bounded above by x 2 + y 2 + z 2 = 4 z and below by x 2 + y 2 = z 2 . 13 Evil Example - If this is clear to you, then you’re awesome. Convert Z π 2 Z √ 3 1 Z √ 4-r 2 1 r sin θ dzdrdθ to spherical coordinates. 14...
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This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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Lecture 13-7 - Defn Spherical Coordinates A point P in...

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