# S 15-4 - Math 200 Spring 2010 Worksheet#20 Name 142 Section...

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Unformatted text preview: Math 200, Spring 2010 Worksheet #20 Name 142 Section 15-4 Inte ration 1n Polar C linderical andS hericalCoordinates 7112 Erdr d 9 Sketch the region whose area is given b39the integral. 1. Evaluate the integral I 6 12' R x2+y2=landx2+y2=4 R :§(Y‘91l)151’52 1% £6£3%Z 37,1.1 511 (W, R‘f {(TC059+Y‘91V\6) Y‘Arde ‘91’23 o X+7v ;g; r .2; fr ((559 +Sm8);“ag: «{(:°SB+S’“ééﬁf:‘ir X1117 5“?)1‘110 3, I :{5m9‘C05912/L- [Air/glﬁ [-1-0-(1—‘011 [341% 3. Use polar coordinates to ﬁnd the volume of the solid that IS below the paraboloid z = 18 — 2;:2 a2 y and above The Port. 130(01J. lhi‘f‘l‘SGCi‘; "HQ )4). -P[Q“e m 'H‘te CNN-If Lurk rad-vs 3 5" 0:) a +2 :IS V \$5,111 % 1X 2 7)d9215[1:13-1(x::f)]cm XX LL“: "'0' 3 ‘f the xy-plane. My i3 X17325 ? 311‘ 3 3 1:; of113-1r1)vdrc\9' = 1221(13r—2r2hr 3 ‘1” 1—Hi-0 ° 21. 910 [qr at 10 ~ 3n‘(31~ 1 e Vlﬁ‘ 4. Use polar coordinates to find the volume of a sphere of radius a. For 0t \$ph€re O'F Tacitus Q (g) Sammie; \Jrzﬂl 121m 211 6“!” X17 lane. X+7¢—Q. 1:1j7WZdrol9: 2E5 IrWdr __ 1:9]:r[3‘( HY]: .13 LP 3 ’rg"-;.1'I".9v - 5. Maﬁa 0 5. Evaluate the integral [01,2 Ema rdrdB. Sketch the region whose area is given by the integral. 5‘94 “H1653 R:\$(rje)lofr£'-lcose)0595%§ WAY-(:19 E 3‘ l < o U394 (use =1ll+€6:29) Va t+£05.63 3. :[ZKJJ]? 1': these 9';- 1 Ab TX:- Lt:c°59 .. + .. 1"] "J, 3035949 =le-(l+cosle)¢la ( :L‘Il‘xd O .J/ 0 X" 3 +7, HLl’ bunch”? ‘5 0561- => 3 _ 4M Susie] 1“:31? 94 7 ‘9 Eagytle 6. Sketch the region of integration and evaluate by changing to polar coordinates. Duadraﬁ't‘ ) WJ’E—z a): C(Rs II x +y dxdy _ “0 0‘ §(X,y)lofin/oixfllq~7L§ '2. 3 _. 1T .1 ‘ (no 03130195 llr rat-ate g f ll J :4 ° ° ~¥ - u. 2_ - x .. % ‘3 '- (/7) X ”l“? V .- jdejrld - [ ')(L J. 3 3 - ‘1‘ C11 0 0 T h 610 3}“ o _. )4 Cl 1 2. w 5; mm a Jlr 2. :erar]::o ll) “ f§ﬂ(e)]49 wkxck agrees w't-HA -er Single Unﬁt-Fable RIMMIL qtor— {-lxe area. 0F k lOolar- rear‘ww 8. Express this integral in cylindrical coordinates. Ila—L4 f( )6va WWW w:(Sﬁgzéndéxflj-lJl-xi5)f l—xz) 05234} 2 [311' I L} PrOJQCl‘lOn o'l: LU owl-o ~er Xy-flmne. 1.5 o oj6/¥{VC058)Y‘SN\Q ﬂraedrde D: §{X X7” —l 4x51 ~x“ 955?"; .lh {)Olar (cardinal-er) g— {[rgﬂo 952T,05r‘flg .ln Calﬁxdhcﬂ Coordmai'es 3 us: §(Y‘ veto/ﬂ} 595 mjofﬁIJOS-ESI“ 9. Evaluate “L (x3 + xy2 )dV , where Wis the solid in the ﬁrst octant that lies beneath the paraboloid z = l — x2 — yz. \inE @armbolmg why-5%}; %e X) 'fIQNQ whet“ 2: 0 => 0" :i‘X' y f? X; +y 9~=l I“ Cailndhcai Cabrdsti—e‘f/ ‘HHS 1-5 I“ -i . 1““?- JA‘e ”if 13 m rue vcii-rt Mint 10— N (ream- e- K Osage 052:1- r} 155/59] {My 3‘0“! ‘Ofo V/f(rmCose-¥rtosesfne)rd&drd9 7x :jOl/o FZC058{(056+81n6)d2dral6:{jfrg C059[2}::;rde %. :j forvl-rUCosédrdé : (aseaa OIW‘ r Ar l O o O .1... 1 - = 5.2ng an: ( )~ 3% 10. Find the volume of the solid that lies within both the cylinder x2 + y2 =1 and the sphere x2 + y2 + 22 — —.4 In Cyll'ndrieul Coorqu'i'es) DJ If 'Hﬁe SOiIJ Nglenz uni-hm ‘Hwe C5, |nci€r r- i and ioowxoleJ helm...) and above by ‘Hu? thf’f’a r 22-4-2 = Li- ) 3° w: §(r"::)562‘0 643T)“ “U ‘“ a-r‘iesmg +1: ” 111M171”? Ydédrda °~1F~TT =5:er 2—HT We :jyhmdrae Ji=rﬁr 3/\ H 21-I-1—rdrl2rril'3H )(w) 0 -3941. ?1 " 33-3) 3Tt'(3"‘ 3) 11 Evaluate ”LN— x2 + y2 + 22 d V where Wis the solid hemisphere that lies above the xy— —plane and has center th (1 d 1. " a eoriginan raius .. §(}D)9CP))O £2051JO£9£21T}U:‘9059{3 SUZWW ffﬂpmﬁf 2, 45 .. j SInCPd/OOIQOJQ x :lrre [6:345 Sihqool‘ﬂjf:6‘1f = 2n I“; (05%???[11230-10' : 2w (11a :1; 12. Find the volume of the smaller wedge cut from a sphere of radius a by two planes that intersect along a diameter at an angle of 7r/ 6 ﬂame 44w 0?!er 0+ 1% Sphere Q1: (0 o, 0). Le‘l‘ 'Hw dr‘amG'i‘er‘ a-P inkrSoc’ir‘am be along HP 511- -ax:s, one cs-Y-ﬁerllaner be -H‘e X2? lean'e and 'Hne o‘er he +keflome whose angle tel-Ht Hie x2 flan-e I5 9“ 1V6. 1 TR“ no“ {the :17)! 03954 ose‘T/é 0 ¢2<r~§ “ _ W6 ad U=£ﬂAUI-‘fhgf/fa31n¢dfd¢de =[de/sntpdCPJdef3 43%”: 035(19): [37;]:— 12: :2 30. = 31%)»3 13. Use spherical coordinates to calculate the triple integralo of f (x, y, z) — -1/x2 + y + z over PE—{(x,y,z).x +y:+z \$22}. 7‘ 3+7 +2 52% Sphere f f Tyros? )‘lﬂlthﬁffl Casgtljrn Slcostﬂ “é E059 (9)105942n awry, ogm Damp} mm av =515yolv =12? 072”” n, ff smwﬁ‘ﬂdg : [23153; }JJ: :wdcm m y, (Jo/Li (as 905m<i9d¢3019= if? “(0590)::11/36 :1 3: T ‘8? 14 Find the volume enclosed by the torus p— - sin at aegis“. EncloSeJ by 'HxP ‘hrus LU: 23(3))6 (ﬁlo: 94:271.} Q4q34r- )O‘f < SUP} 211‘)“- Suﬁp TI" 71‘ - ' U:5ﬂJU:Mj/ij01§lh¢dfdcpde— jefsmcp[;j?;_:“\$ _-£,_-. 0 6 f—"o a 3,, 7‘? 'L X 3; fﬂtaﬂfﬁong) c149 O .. 2’11" BojSm ”(Odtﬂ §Hf[l ~10le :q‘J: - gnfﬁﬁ i i - 3 ~ @3ch 4 Cos 3 2 . 7* a qcﬂ)°lqj: 3T? [BQ‘IL‘SInQCpTﬁSIAHQ] :_'%ﬁ'1'ﬁ’ [1.35ch talt’n'h‘hQs Smax= M Me) (05;: 3 H'WQXJ o - __L 8 2. Homework #20 Reread Section 15. 4 l ‘4‘ 1T (Due 04/07) Do Section 15.4 Exercises: 1,3, 7, l3, l5, 17, 21, 30, 31, 33, 39, 41, 45, 48, 51, 55, 58, 61, 67 Prepare for next class session: Read Section 15.5 ...
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