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notes-1 - TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL...

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TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES DAVID ANDREW SMITH Abstract. In this topic we state the usual definition of cylindrical coordinates and work out examples of converting Cartesian integrals to triple integrals in cylindrical coordinates. 1. Triple Integrals in Cylindrical and Spherical Coordinates 1.1. Triple Integrals in Cylindrical Coordinates. Each point in three dimensions is uniquely represented in cylindrical coordinates by ( r, θ, z ) using 0 r < , 0 θ < 2 π, and -∞ < z < + . The conversion formulas involving rectangular coordinates ( x, y, z ) and cylindrical coordinates ( r, θ, z ) are r = p x 2 + y 2 tan θ = y x z = z x = r cos θ y = r sin θ z = z. A triple integral Z Z Z R f ( x, y, z ) dV can sometimes be evaluated by transforming to cylindrical coordinates if the region of integration R is z -simple and the projection of R onto the xy -plane is a region D that can be described naturally in terms of polar coordinates. Theorem 1. Assume R is a solid region with continuous upper surface z = v ( r, θ ) and continuous lower surface z = u ( r, θ ) and assume is D be the projection of the solid onto the xy -plane expressed in polar coordinates: D = { ( r, θ ) : α r β, r 1 ( θ ) r ( θ ) r 2 ( θ ) } where g 1 and g 2 are continuous functions of θ . If f ( x, y, z ) is continuous on R, then the triple integral of f over R can be evaluated as follows Z Z Z R f ( x, y, z ) dV = Z β α Z r 2 ( θ ) r 1 ( θ ) Z v ( r,θ ) u ( r,θ ) f ( r cos θ, r sin θ, z ) rdzdrdθ. Example 1. Use cylindrical coordinates to find the volume of the solid bounded by the paraboloid 4 x 2 + 4 y 2 + z = 1 and the xy -plane. Solution. The projection of the solid region onto the xy -plane is the region enclosed by x 2 + y 2 = 1 4 , see Figure (1). In cylindrical coordinates the volume is determined as, V = 4 Z π/ 2 0 Z 1 / 2 0 Z 1 - 4 r 2 0 rdzdrdθ Date : February 24, 2011. 1
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2 DAVID ANDREW SMITH - 1.0 - 0.5 0.0 0.5 1.0 - 1.0 - 0.5 0.0 0.5 1.0 0.0 0.5 1.0 - 0.4 - 0.2 0.2 0.4 - 0.4 - 0.2 0.2 0.4 Figure 1. the paraboloid 4 x 2 + 4 y 2 + z = 1 bounded by the xy -plane = 4 Z π/ 2 0 Z 1 / 2 0 ( r - 4 r 3 ) drdθ = 4 Z π/ 2 0 1 16 = π 8 .
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