Copyright © Cengage Learning. All rights reserved. 15 Multiple IntegralsCopyright © Cengage Learning. All rights reserved. 15.8 Triple Integrals in Cylindrical Coordinates3 Cylindrical Coordinates Learning objectives: § convert between rectangular and cylindrical coordinates § evaluate triple integrals with cylindrical coordinates4 Cylindrical Coordinates5 Cylindrical Coordinates In the cylindrical coordinate system, a point P in three-dimensional space is represented by the ordered triple (r, θ, z) where r and θ 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.) Figure 2 The cylindrical coordinates of a point6 Cylindrical Coordinates To convert from cylindrical to rectangular coordinates, we use the equations whereas to convert from rectangular to cylindrical coordinates, we use7 Evaluating Triple Integrals with Cylindrical Coordinates8 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 69 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, θ ) | α ≤ θ ≤ β , h1(θ ) ≤ r ≤ h2(θ )} We know10 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.11 Evaluating Triple Integrals with Cylindrical Coordinates It says that we convert a triple integral from rectangular to cylindrical coordinates by writing x = r cos θ, y = r sin θ, leaving z as it is, using the appropriate limits of integration for z, r, and θ, and replacing dV by r dz dr dθ. (Figure 7 shows how to remember this.) Figure 7 Volume element in cylindrical coordinates: dV = r dz dr dθ12 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
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