15 Multiple Integrals Copyright Cengage Learning All rights reserved 15 8 Triple Integrals in Cylindrical Coordinates Copyright Cengage Learning All rights reserved Cylindrical Coordinates Learning objectives convert between rectangular and cylindrical coordinates evaluate triple integrals with cylindrical coordinates 3 Cylindrical Coordinates 4 Cylindrical Coordinates In the cylindrical coordinate system a point P in three dimensional space is represented by the ordered triple r z where r and are polar coordinates of the projection of P onto the xy plane and z is the directed distance from the xy plane to P See Figure 2 The cylindrical coordinates of a point Figure 2 5 Cylindrical Coordinates To convert from cylindrical to rectangular coordinates we use the equations whereas to convert from rectangular to cylindrical coordinates we use 6 Evaluating Triple Integrals with Cylindrical Coordinates 7 Evaluating Triple Integrals with Cylindrical Coordinates Suppose that E is a type 1 region whose projection D onto the xy plane is conveniently described in polar coordinates see Figure 6 Figure 6 8 Evaluating Triple Integrals with Cylindrical Coordinates 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 h1 r h2 We know 9 Evaluating Triple Integrals with Cylindrical Coordinates But to evaluate double integrals in polar coordinates we have the formula Formula 4 is the formula for triple integration in cylindrical coordinates 10 Evaluating Triple Integrals with Cylindrical Coordinates 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 and replacing dV by r dz dr d Figure 7 shows how to remember this Volume element in cylindrical coordinates dV r dz dr d Figure 7 11 Evaluating Triple Integrals with Cylindrical Coordinates It is worthwhile to use this formula when E is a solid region easily described in cylindrical coordinates and especially when the function f x y z involves the expression x2 y2 It is your job to decide whether to use rectangular or cylindrical or spherical coordinates to evaluate a given triple integral similar to the choice between rectangular and polar coordinates to evaluate a given double integral 12
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