Section 15 7 Triple integrals in cylindrical coordinates MATH 251 Fall 2023 In the cylindrical coordinate system a point P x y z is represented by the order 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 P to the xy plane Convert from cylindrical to rectangular coordinates Convert from rectangular to cylindrical coordinates x r cos y r sin z z r2 x2 y2 tan z z y x Example 1 1 Find the rectangular coordinates of the point with cylindrical coordinates 1 6 3 2 Find cylindrical coordinates of the point with rectangular coordinates 1 1 5 1 2 Example 2 Write the equation in cylindrical coordinates 1 x2 y2 5 2 z x2 y2 3 z x2 y2 4 x2 2x y2 2y z2 2 Triple integrals with cylindrical coordinates Suppose that E is a type 1 region whose projection D onto the xy plane is described in polar coordinates i e 3 E x y z x y 2 D u1 x y z u2 x y D r h1 r h2 f x y z dz dA from Section 15 6 We have E u1 x y y D Z u2 x y f x y z x r h1 Z u2 r cos r sin Z r h2 Z 3R p9 x2 Example 3 EvaluateR 3 p1 x2 R x2 y2 u1 r cos r sin zdzdydx 0 The above formula is the formula for triple integration in cylindrical coordinates f r cos r sin z dz rdrd from Section 15 3 4 Example 4 Evaluate tE ydV where E is the solid bounded by the paraboloids z x2 y2 z 50 x2 y2 and the half space of positive y values Example 5 Set up but do not evaluate the integral tE x2ydV where E is enclosed by the cone z p4x2 4y2 the plane z 0 and the cylinder x2 y2 16 5 Example 6 Set up but do not evaluate the integral tE xydV where E is enclosed by the cylinder x2 y2 4 and the sphere x2 y2 z2 16 and contains the origin
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