2224-Sec13_7-HWT

2224-Sec13_7-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus–Sec.13.7: Triple Integrals in Cylindrical and Spherical Coordinates I. Integration in Cylindrical Coordinates Cylindrical Coordinates are obtained by combining polar coordinates in the xy -plane with the usual z-axis. A. Definition Cylindrical coordinates represent a point P in space by ordered triples ( r, θ ,z ) in which 1. r and are polar coordinates for the vertical projection of P on the xy -plane 2. z is the rectangular vertical coordinate. B. Equations Relating Rectangular and Cylindrical Coordinates x = r cos ! , y = r sin , z = z x 2 + y 2 = r 2 , tan = y x C. Descriptions of r=a, = 0 , z=z 0 1. r=a describes a cylinder of radius a about the z –axis 2. = 0 describes a plane that contains the z –axis and makes an angle 0 with the positive x -axis. 3. z=z 0 describes a plane perpendicular to the z –axis D. Integrals in Cylindrical Form 1. Volume of a wedge: ! V = ! z r ! r ! " 2. Volume:
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This note was uploaded on 04/05/2008 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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2224-Sec13_7-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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