18 02A Topic 42 Triple integrals rectangular and cylindrical coordinates Author Jeremy Orloff Read SN I 3 TB 20 5 20 6 Ingredients w f x y z a funcion of 3 variables D a volume in space Examples Box Cylinder Sphere z The triple integral is the usual sum over all the volume elements making up D ZZZ f x y z dV where dV dx dy dz dy D x Even more than in two dimensions the hard part is finding the limits of integration The rest of the hour will be examples Example 1 Find the average distance of a point in the tetrahedron ZZZ 1 shown to the xy plane i e compute Ave f x y z dV V D Limits we ll use vertical line segments to fill out the volume z goes from 0 to the slanted plane x y are in R the projection of D to the xy plane inner variable z 0 to 1 x y middle variable y 0 to 1 x R outer variable x 0 to 1 f x y z distance to xy plane z Z Volume of tetrahedron 1 6 ave 6 z dz dy dx 0 z2 Inner 2 Z Middle 1 x y 0 1 x 0 0 0 1 x y 2 2 1 x y 3 1 x y 2 dy 2 6 1 x 4 Outer 6 24 1 0 1 4 1 1 x 0 1 x 3 6 y z 1 1 y 1 x R y 1 R x y 1 1 1 Z 1 x Z 1 x y dz dx x 18 02A topic 42 2 Example 2 For a cylinder of height h radius a and density 1 find the moment of inertia about the central axis RRR 2 RRR 2 I r dm D D r dV Limits Inner z 0 to h x y in R which is a disk of radius a Inner z 0 to h Middle r 0 to a Outer 0 to 2 Z 2 Z a Z h r2 dz r dr d I z 0 0 Inner integral 0 h 3 r z0 Z a Middle integral x z hr3 r3 h dr 0 Outer integral 2 y a4 h 4 a4 a2 h M 4 2 Example 3 Find the volume between the paraboloid z x2 y 2 and the plane z 2y How to find R Pictures if possible sometimes need algebra C 0 projection of intersection of the surfaces That is C 0 is given by x2 y 2 2y x2 y 1 2 1 a circle Limits inner z from x2 y 2 to 2y the yz section shown at right shows the plane is above the parabola middle r from 0 to 2 sin outer from 0 to Since we re using polar coordinates we need to rephrase the inner limits Inner z from r2 to 2r sin Z Z 2 sin Z 2r sin V r dz dr d 2 0 0 r2 R y C0 x Example 3 graph z z 2y y Example 3 section in yz plane y We call x y z rectangular coordinates We call r z cylindrical coordinates 1 C0 z z dx Example 3 curve C 0 dz dz dy x x dr r d y dy dx Vol element in rectangular coord x z y2 y d Vol element in cylindrical coord 18 02A topic 42 3 Example 4 Find the volume between the paraboloids z 4 x2 y 2 and z x2 y 2 z R projection of volume to the xy plane C 0 projection of the intersection of the surfaces i e x2 y 2 4 x2 y 2 x2 y 2 2 z 4 x2 y 2 0 That is C is a circle of radius 2 centered at 0 Limits inner z x2 y 2 to 4 x2 y 2 r2 to 4 r2 z x2 y 2 middle r 0 to 2 y outer 0 to 2 Z 2 Z 2 Z 4 r2 R C1 r dz dr d Easy to com Volume x 2 0 0 r Part of volume in first octant pute Example 5 Use cylindrical coordinates to find the center of mass of the hemisphere shown Assume 1 By symmetry it s clear xcm 0 and ycm 0 ZZZ ZZZ 1 1 zcm z dm z dV M M D D 16 Clearly R is a disc of radius 2 and M 3 p Limits inner z from 0 to 4 x2 y 2 4 r2 z middle r from 0 to 2 2 outer from 0 to 2 Z 2 Z 2 Z 4 r2 3 zcm zr dz dr d 16 0 0 0 4 r2 3 4 r2 3 z2r 3 4r r3 Inner r 16 2 0 16 2 16 2 2 3 r4 3 Middle r2 16 8 0 8 3 3 3 2 zcm Outer 8 4 4 End of topic 42 notes y x x2 y 2 z 2 4
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