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Unformatted text preview: CachC10_4notes 10.4 Area & Lengths in Polar Coordinates
WarmUp The length. of the curve determined by x = 32‘ and y =
from t: 0 to t: 9 is A) Lﬂ/Qtﬁztﬁdt Bj [:62 9—416ridr
0L: x/9+16t d: D) J03J9~16r2dt
E)! «/9+16t A Elk ,) CH m r." we. ( At 3 A: ll
“’1
SJ 0} Nee?“ CW
0 We want to find the area
inside a length of are. We cut the length into little
pieces and add them. “54’ Rel/mean bore? length: I dx2 + dyg— 2d): dx dt =I 1+[%]2dx :J‘ [%]2+[%]Zdt CachC10m4notes Nowwe want d6: 1:! 63,362+ dyz d6 3:: recs!) y=rsin19 :'—‘=—rsin19+%cosc9 (Wadim 11113) » [m—~rsinﬂ+—cosﬂ]
air [d—6]=r sin 19— —2‘sEn8cosf)%+ [d0] cos 8 £11: rc056+ gaina I
[%=rcos€+:—;sin8] [5%]! =r cos B+2rsi1n9cos€%+[:—;]z sin .9 Wawantdﬂ
I I
add 3 + 35!.
d3 d3
_ I
r’sh’ﬂ—erinﬂcosHj—g+[§L] 1:051!)
1 1 . dr 1 . 3
+ 1" cos 3+2rsu13c050—+ — sin 9
d9
1 I
rls‘zn‘ﬁH rimszﬂi £ sin'6‘+ i @0520
d1? d9
H(sin’3+ cm’B) + d—r I{Slinléihzns’ii'}
d3 dr 2 #1:} {a} (I)
1:; r1+[§;]1 d8 CachC10_4notes >< .: VC 03 B 7:: \l rs”; (.3
ex. r=sin2£9 0:16:2
"a1: 2cos 26’
d6 1:!”14 /sin2 29+ 4003226 (276? Calculator does not need % _
0 . . to be in polar form Area Remember how to ﬁnd the area of a sector: CachC10_4notes p. 674 2. Find the/aﬁ[email protected] the region that is bounded by the given curve
and lies Me speciﬁed sector.
r=e§f2 @59327r 2 31?
”ﬁg (6%) AB 3‘ ”Jig @949 Cu Q‘QS LI 4% m0)"
8. Find the aiegfof the ﬁladed region. A; ﬁg (Smblojzd B r : sin 46? ‘4 '" “‘2; TT .” “LEM (a)
M Q)» ”a 25(p.1’16q {36f .’ {Vi—A" u “h“ _CachC10_4notes l3. Sketch the curve and ﬁnd the area that it encloses. 7'24—sin9
, 1r?
/ w ‘5 "gm (Ll“gm e) A (9*
_ ”agrees — A q 0
Sumo 22. Find the area of the region enclosed by one loop of the curve. 7' 2 2+ 30036 M,_,_ > ( ' 2+5Lus 6) 1 O lnner ‘06? C058; ”3:; CachC10_4notes I 28. Find the area of the region that lies inside the ﬁrst curve and
outside the second curve. ( ' ”“31 F: iii” (9
I
I
O: H‘L‘NG O I 30.35 0
“.1 [email protected] Circa56
9v'ﬁ
a. ”$13"?
“‘T It
\ 2, l 2 (a 9316‘ g
2 Z (+L0§®)J0 h“ 2 TV C03 38. Find all points of intersection of the given curves. r=2, . r=200326 '2 1: ZCQS ZG \ I: CoQQ 2'9; 0 J 2'“) LtTi”) L.«— .4 2:;‘1952—6 8: (3)113 2n).._ W‘ ““29" 29 '3 “WI 311" 911" ,,
} ) 931?? 33;? 53%“
[2,11’) (:2 3F ’ 2*) ZJ‘ CachC10_4notes I 91;" w
ow .1 «3g 1 1,, 1
47. Find the length of the polar curve.
01.,"
r=235 0S6S2ﬂ,‘ A9'5’rlnlw \ 3 L1 , Ci 5 H
dye
51. Use a calculatOI 01 compute1 to ﬁnd the length of the loop i f” G ‘‘‘ ““" 2 $ \ A 16)
oouect to four decimal places. M»
/
One loop of the four—leaved rose 7‘ : cos 2&9 ”
(9 2 rr 4T “1:
0 4 /_ TT .  I . , “a
_ 8 S' \chm'a.,_€ﬁl 1+ (—25%.«263 dg ...
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 Fall '12
 Wade
 Calculus, Polar Coordinates

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