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This 5 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 2433 at University of Houston taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/208411/math-2433-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
SUMMARY CHAPTERS 16 and 17 CHAPTER 16 MULTIPLE INTEGRALS I DOUBLE INTEGRALS a De nition Let f fXy be continuous on the rectangle R a g X g b c g y g 1 Let 73 be a partition of R and let m and IVA7 be the minimum and maximum values of f on the Lj suberectangle R Then 1 Lower sum Lf73 szi39 Xi M 113 Upper sum Uf73 ZZMj Xi M M 111 Riemann sum Sf73 fXy X1 yj Where ngg isa point in R i1 391 u The double integral of f over R is the unique number I that satis es Lf73 g I g Uf73 for all partitions 73 Notation I fXydXdy R Let be an arbitrary closed bounded region in the plane Then 9 fXydXdyRFXy dXdy Where R is a rectangle that contains and F X fX y on and F X y 0 on R7 b Repeated Integrals If the region is given by a g X b 1X g yg 2X Type I region then b 152m fXy dXdy fXy 1de 9 a 151m If the region is given by c g yg d 10 g X 20 Type II region then 11 2y fX y dXdy fX y dXdy 9 0 1PM c Polar Coordinates fXydXdy fr cos rsin rdrd r2 r2 1 Applications 13 11 111 IV Volume lf fXy Z 0 on then V fXy dXdy is the volume of the 9 solid cylinder that has the surface Z fX y as its top sides as its base and vertical Area 1 dXdy area of 9 Mass of a Plate If the density of a plate at a point Xy in the closed bounded region is given by a continuous function X then the mass of the plate is M X y dXdy 9 Center of Mass of a Plate Let the continuous function X be the density function of a plate Then the coordinates XM yM of the center of mass of the plate are given by QX XydXdy Qy XydXdy T yMT Where M is the mass of the plate XM 11 TRIPLE INTEGRALS a De nition Let f fX y Z be continuous on the box T a1 X az b1 y bz 012 02 Let 73 be a partition of T and let mijk and lm be the minimum and maximum values of f on the jjk subebox Rigk Then n m l 1 Lower sum LAP mm Xi yj Zk i1j1k1 n m l 1139 Upper sum Uf73 2 Mjk Xi yj Zk i1g1k1 n m l 111 Riemann sum SAP fXyfk Xi y Zk Where 4922 is i1j1k1 a point in Rigk The t ple integral of f over T is the unique number I that satis es LAP S S Uf73 Notation I fX y Z dXdde T Let be an arbitrary closed bounded region in space Then fXyZdXdyZAFXyZdXdde and FXy Z fXy Z on for all partitions 73 Where T is a box that contains and F X y 0 onTi b Repeated Integrals If the region is given by altxlt b 1ltXgt y zoo my 2 my Type I region then b 152w 2y fXy Z dXdde fX y Z dz 1de Q a 1w INN Note There are ve more types of special regions c Cylindrical Coordinates QfXyZdXddeAfTCOS rsin Zrdrd dz CHAPTER 17 LINE INTEGRALS Given a vector eld FXy PXy 391 QXyJ39 and a smooth or piecewise smooth curve C C X Xu y yu a g u g b parametric form C ru Xu i yuj a lt u g b vector form Or in three dimensions a vector eld FX y Z PXy Z i QX y Zj RXy Z bfk and a smooth or piecewise smooth curve C C X Xu y yu Z 2a a g u g b parametric form C ru Xu i yuj Zu k a g u g b vector form DEFINITION The line integral of F over C is the number given by b Fra r Frur udu C a Alternative notations Fr a r PXy dX QXy dy FT d5 2 dimensions 0 C C 01quot Fr dr PXy Z dX QX y Z dy RXy Z dz FT d5 3 dimensions 0 C C Where FT is the component of F on the unit tangent vector T THEOREM Line integrals are invariant under orientationepreserving changes of parameter THEOREM Reversing the orientation of C changes the sign of the integral LCFra r7CFrdr cf fX 1X 7 fX dX FUNDAMENTAL THEOREM OF LINE INTEGRALS Given a curve C ru a g u g b and a vector eld F If F Vf for some function fX then C Fr a r fB 7 fA Where A ra and B rb DEFINITION The curve C is closed if ra rb COROLLARY 1 If F is the gradient of some function f and the curve C is closed then Fr a r o C COROLLARY 2 Independence of Path If F is the gradient of some function f and if C1 and C2 are any two curves Which begin at A and end at B then ClFra rC2Frdr DEFINITION The curve C is a simple closed curve if it is closed and rt1 7 rt2 for all a lt Q lt Q lt b The positive direction on C is counterclockwise A simple closed curve is also called a Jordan curve The region enclosed by a simple closed curve is called a Jordan region GREEN S THEOREM Given a simple closed curve C oriented in the counterclockwise direction and a vector eld FXy PXy i QXyj PXydXQXJydyQ7Qiil dXdy Where is the Jordan region enclosed by C COROLLARY TO GREEN S THEOREM If C is a simple closed curve and if 7 70 that is if F PXy i QXyj is a gradient then PXydXQXdy0 AREA OF USING GREEN S THEOREM C a simple closed curve enclosing the region Area of ldXdyf idef Xdy ideXdy o c c 2 c
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