18 02 Multivariable Calculus Spring 2009 Lecture 29 Cylindrical and Spherical Coordinates and applications April 17 Reading Material From Simmons 18 7 20 6 and 20 7 Last time Triple Integrals Cylindrical Coordinates Today Cylindrical and Spherical Coordinates Applications 2 Cylindrical Coordinates Cylindrical coordinates are obtained by using polar coordinates in the xy plane and the z coordinate along the z axis to summarize rectangular x y z cylindrical polar z r z Conversion Like polar r z x y z x r cos y r sin z z r z x y z p r x2 y 2 tan 1 y x z z Integration in Cylindrical Coordinates dV ZZZ ZZZ z f r cos x r sin x z r dz dr d f x y z dV D r z limits 1 Question Why dV r dz dr d 2 1 If one wiggles r z how big is resulting cube From the picture is then clear that V r r z hence if we pass to differentials we have dV r dz z dr d typical order which makes sense since we have to take into account the rate of change of the area of the base with respect to and r and the rate of change of the hight with respect to z Question When do we use cylindrical coordinates They are very good when the shadow has rotate symmetry Exercise 1 Set up cylindrical integral Find the mass of the part of paraboloid D described by z 4 x2 y 2 on the upper half space and of density x y z x y z Solution The shadow of the paraboloid is the disk centered at the origin of radius 2 This is a perfect region to describe in polar coordinates hence we should use the cylindrical coordinates to write the triple integral representing the mass 2 3 Spherical Coordinates These coordinates are good for pieces of spheres hemispheres ice cream cones Description of spherical coordinates Note that here 0 2 0 and Conversion r z x y z r sin z cos x r cos sin cos y r sin sin sin z z cos Integration in spherical coordinate Given a region D in space 3 1 Find limits Z Z Z z z from shadow from sheet at angle 2 dV 2 sin d d d then we have ZZZ Z Z Z f x y z dV D f sin cos sin sin cos 2 sin d d d Question Why dV 2 sin d d d If one wiggles how big is resulting cube From the picture is then clear that V sin hence if we pass to differentials we have what we wanted Exercise 2 Compute the volume of sphere of radius a Solution We need to compute the triple integral ZZZ dV volume sphere 4 3 1 Obviously the best coordinate system is the spherical one 4 The Jacobian Following the 2D model of change of variables we will call Jacobian of the cylindrical change of variable the determinant of the matrix x y z x r y r z r x z y z x z r sin r cos 0 cos sin 0 0 0 1 r hence dV r d dr dz as in 2 1 On the other hand for the spherical change of variable the Jacobian is given by x y z x y z x y x sin sin sin cos 0 sin cos sin sin cos cos cos cos sin sin 2 sin hence in this case dV 2 sin d d d as in 3 1 5 Applications Given a solid D in space of density x y z we already computed the Mass of D as Z Z Z Mass D x y z dV D Now if f x y z is a function defined on D we define the Average of f to be RRR f x y z x y z dV D Average f Mass D 5 The Center of Mass of a solid D is the point P x y z where x is the average of the function x y is the average of the function y and z is the average of the function z Finally we consider the Moment of Inertia We have The moment of inertia of a solid D of density x y z about the z axis Z Z Z Iz x2 y 2 x y z dV D The moment of inertia about the x axis Z Z Z Ix z 2 y 2 x y z dV D The moment of inertia about the x axis Z Z Z Iy z 2 x2 x y z dV D Exercise 3 Compute the moment of inertia Iz of the solid cone R between z ar and z b Here assume 1 6 Solution 7 Study Guide 1 Consider the following questions Consider the cylinder with a disk of radius a as a base and hight h Using the formula for the volume given by cylindrical coordinates and by fixing r a in this formula compute the area of the lateral surface of the cylinder Hint Note that there is no variation dr in this case Consider the sphere of radius a Using the formula for the volume given by spherical coordinates and by fixing a in this formula compute the area of the sphere Hint Note that there is no variation d in this case 8
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