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15.4withScannedExamples

# 15.4withScannedExamples - 15.4 DOUBLE INTEGRALS IN POLAR...

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Unformatted text preview: 15.4: DOUBLE INTEGRALS IN POLAR COORDINATES KIAM HEONG KWA 1. DOUBLE INTEGRALS IN POLAR COORDINATES Recall that the polar coordinates (r, 0) of a point in the xy—plane are related to the rectangular coordinates (22,31) by the equations 7‘2 =xz+y2, Izrcosﬁ, y=rsin0. Convention. The notation (7‘, 0) always denotes the polar coordinates of a point in this section. By a polar rectangle is meant a domain in the xy-plane that has the form R:{(7,0)ER2IGST‘Sb,aSQSﬂ}’ where 0 g a S b < 00 and 0 S [3 —a < 2%. For a continuous function f over the polar rectangle R, we would like to deﬁne the double integral of f over R in terms of polar coordinates. We do this by introducing the double Riemann sum in terms of polar coordinates. To begin with, we divide the interval [a, b] into m subintervals [n-1, n] of width b—a m Ti—T1;_1=AT= ,i=1,2,...,m, and the interval [a, ﬁ] into n subintervals [0j_1, 01-] of width B—oz n 9j—0j_1=A0= ,j=1,2,"',’l’L. This process results in mn non-overlapping polar rectangles RU con— tained in R. See ﬁgure 4 on p. 974 of the text. More precisely, we have R’ij = {(736) E R2l’r‘i—l S T S. riyaj—l S 6 S 6j}? i=1,2,--- ,m, j=1,2,--- ,n. Date: October 13, 2010. 2 KIAM HEONG KWA Each of these non—overlapping polar rectangles has an area AAU de— pending on its indices, i.e., l 1 AAiJ: {391 —9j— 1)— 2T-2_ 1(91—91—1) 1 = 5022 — 7‘12—1)(9j .— 0j_1) l = 502‘ +Ti—1)(7“i — 72—le — 91—1) = rfArAO, where we have written 7‘: for (rt--1 + n) / 2. Also, denoting (0j_1 + 6]) / 2 by 0;, we see that the rectangular coordinates of the center of Rij are given by * * * ‘ * (n- cos 0]. , Ti sm 9].). We deﬁne 221% r: cos 9;,7": sin 9*)AAij i=1j=—1 =ZZf( 7" *cosﬁ’l‘, r *sin0*) fArAQ i=1j=l as a double Riemann sum of f over R in terms of polar coordinates. Note that if f is positive over R, then the Riemann sum is approximat— ing the volume of the solid that lies under the graph of f and above the polar rectangle R. Finally, we deﬁne the double integral of f over R in polar coordinates as m n // f( (x y)dA= lim ZZﬂ r *cosH*, 7" *sin6") fArAQ i=1j=l provided the limit exists. Note that we have used the same notation for the double integral deﬁned in polar coordinates as well as the one deﬁned in terms of rectangular coordinates. This shall create no con- fusion for the class of continuous functions since, as a consequence of a more general theory that we shall discuss in section 15.9, if f is con- tinuous, then the double integrals deﬁned in terms of both coordinate systems coincide. 15.4: DOUBLE INTEGRALS IN POLAR COORDINATES 3 On the other hand, note that if we set 9(7‘, 6) = f(7" cos 0, min 0) - 7" for all (1", 0) E R, then we have 22m 7139;) )Am0=::f( 'r *c056*, r *sin6*)r :Arna. i=1 j: —1 i=1 j=1 Therefore [/12 g(r,6)dA=mlrilr_r)10°ZZg(r* ,, 6j)ArA0 i=1j=1 = lim E in f( 7* *cosQ;,r,* sin6*)r :A’I‘AQ m,n—)oo i=1 j: —1 = / R f(x,y)dA, where we have treated g as a function on the rectangle [a, b] x [a, ,6], a rectangle in the rectangular coordinates. In particular, if f is continu— ous, so that so is 9, we have f/R f(:c,y)dA=//Rg(r,0)dA =/: /abg(r,0)drd6 3 b =/ / f(rcos<9,rsin0)'rd’rd0. More generally, if f is continuous on a polar region of the form D = {(7:0) 6 R2lh1(6) s r s h2(6),a s 6 3-6}, where hl and hg are continous functions such that 0 3 MW) 3 h2(6) < ooandOﬁﬁ—a<27r, then h2(9) (1.2) //Df (:r ,y) )=dA /B / f(7"c039,'rsin9)rdr d6. h1(9) In particular, the area of the region D is given by 3 112(9) = / / 7" d?" d0. (1 h1(0) (’(") 21-? 15-" y 20 t; [XX dA —’ f K dA ogre)” offs-9.058 0.405% ’~ _ 0:951}: 5'} _ 9'2“. r=1 r—zwsﬁ “=3 r-Cose rekds“ grhsG ~°W W r=0 f— N 93° 1”: ‘ x*+«ﬂz*21=vs, zzo __P_-;_3_ //’ /_/,-~; / / C 2:: !\¢~KZ‘L’L ) [GVMmAQ-Hmtiij Cmuemtrj Am Poxarcmﬁum P- 4 1 2. \ V —= lX‘X (ix—r? J gawk”: o '1 2 g1+CX~0 ‘\ “X,%:O 1 z 1?}: 1X“?< [#28038 r Q r‘c—OSB -— (716.318 rTLCCp<28+ﬁir€Le>= 9-” “fer r‘ "a 26D50' ...
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15.4withScannedExamples - 15.4 DOUBLE INTEGRALS IN POLAR...

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