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# VECTOR CALCULUS I MTH 254

OSU

GPA 3.79

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This 5 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 254 at Oregon State University taught by B. Peterson in Fall. Since its upload, it has received 13 views. For similar materials see /class/224438/mth-254-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

Double and Triple Integrals Mth 254 Fall 1999 Bent E Petersen Filename doubleintegralsmws This worksheet illustrates Maple s limited support for double and triple integrals You can get Maple to do quite a bit but you will need to understand what you are doing and you may need to do some of the work for example a change of variable Maple support for double and triple integrals via iterated integrals is part of the student package so we need to load that package rst gt restart gt with student D Diff Doubleint Int Limit Lineint Product Sum Tripleint changevar combine completesquare distance equate extrema integrand intercept intparts isolate leftbox 1111 111 leftsum makeproc marimi e midpoint minimi e powsubs rightbox rightsum showtangent simpson slope summand trapezoid value gt with plots needed for implicitplot below Let us look at the evaluation of a simple double integral Note it is probably easier to do it by hand but this example illustrates some of what you will need to go through in the general case Example Evaluate the double integral of xy over the region bounded by the line y x l and the parabola 2 y 2 x 6 First we nd the points of intersection gt solveyx 1 yquot22x6 XY x1y2y4x5 Now we plot the curves over a slightly larger range experiment gt implicitplot yx 1 yquot22x6 x 4 6 y 3 5 thickness2 Page 1 xsthewaytogo h r us mm Thatxspretty m gt glsolve 22txax gt 92 solveyxe1x g2 y1 gt Doubleintxyxql gzyez 4 95 value s 1 ff xydxdy36 2 m2 w evaluate the double integral When we do mterated emeng by handwe sometimes nd n eonvemem or even necessary to change m Mm value is dif cult to interpret e for example 95 gt Doubleintx 3expx 5 Fy 1yo 1 hs95evalfhs9623 Pagel 11 5 ij3exdxdy 0 y 4 45 4 45 4 4 1Fg115 y5 F1115 y5 y4F y4fg y5 1 5 45 ys 1 1 5 If e dx dy 3436563656918090470720574943 0 y Note when we apply evalf to a Doubleint Maple will employ a numeric quadrature method Thus we can almost always obtain an approximate numeric value If we were doing this integral by hand we would have changed the order of integration first Let s try that here gt Doubleintxquot3expxquot5 y0 xx0 1 value lhs evalflhs 28 1 1 1 5 fo3exdydx e 5 5 0 0 1 5 J ff e dy dx 3436563656918090470720574943 0 0 Obviously Maple found the change of order of integration helpful too Example Now let39s consider a triple integral and a famous result of Isaac Newton For a ball B of radius a with density 5 the total mass M is given by gt Tripleintdeltaxyz xyzB JJJ5xyz dxdydz B Let s place the ball so it s centered at the origin and let39s suppose the density 5 depends only on the distance p from the center Then in spherical coordinates 9 longitude I colatitude we have Page 3 gt Tripleint delta rho rhoquot2sin phi rho0 a phi0 Pi theta Pi Pi value Mrhs Jampzsinwwpdwe 77 0 0 JHampzsinwwpade4J6ppzdpn 77 0 0 0 M4Jq5ppzdpn 0 The zcomponent of the gravitational force of the ball on a point mass of mass m located at the point 00s where a lt s is given by gt Tripleint Gmdelta rho rhoquot2sin phi s rhocos phi squot2 2 srhocos phi rhoquot2 A 32 phi0 Pi theta Pi Pi rho0 a value F rhs 7r 7 Gm6p pzsin s pcosw 3 2 d d6dp 2 2 s 2spcosp 0 770 1177 Gm6p pzsinwxs p cos 32 d dedp sz Zspcos pz 0770 2 Gm xpnfmp sz p p s ZSVPS2 p ltpszsndp sMps2 p sz 26m6pp2 p szp p s zs ps2 p pszsndp s2 rsZ gt sz F 0 gt FsimplifyF an 59 p2csgnpScsgnpScsgnpscsgnpSdp F2 S2 i We would like to get lid of the complex signum functions We should be able to do so by making use of Page 4 the assume facility but it seems tricky We will save ourselves a headache by making substitutions for the troublesome expressions since we know their values We are making a simple syntactic substitution here an expression is replaced by another expression with no reference to the value of either one so care is needed Actually to be honest it39s a hack gt Fsubs csgn srho 1 csgnrhos1 F Gmnf Z 5p pzdp 0 F2 S2 Because M already has a value assigned to it we have to unassign it in order to subtitute it into the formula for F It is easy to con lse Maple here by doing something nonsensical so we have to be careful Next to use F in a formula we also need to unevaluate it else it will be replaced by its value gt NM M M simplifyFMN F F F GmM Sz F We have derived Newton s result that a ball with density depending only on the distance from the center attracts a point mass as if the mass of the ball were concentrated at the center Page 5

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