Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Tes solion:. a) Find he cener and radis of he sphere 4 4 4z 8 6 0 z ( ) ( ) z / 4 The cener is a (, -, 0), and radis b) Find he cener and radis of he sphere 4 4 4z 8 6 0. 4 / 4 0 4 4 4z 4 6 8z 44 0. 4 4 4z 4 6 8z 44 0 z z The cener is a (, -, ), and radis 5 6 4 0 ( ) ( ) ( z ) 5. a) Given a vecor r 4,4, he given vecor.. Find a vecor of magnide 0 in he direcion of 0 4 4,4, b) Find and componens of a vecor of magnide 8 and an angle of wih he posiive direcion. 8cos 4, 8sin 4 4. a) Tes if he following vecors are coplanar: a,4, 7, b,, 4 and c 0, 9,8. Use he concep of scalar riple prodc. 4 7 a ( b c ) 4 0, es coplanar 0 9 8 b) Tes if he following vecors are coplanar: i 4 j 7 k, v i j 4k and w 9 j 8k. Use he concep of scalar riple prodc.
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz 4 7 ( v w) 4 0, es coplanar 0 9 8 4. Find orhogonal projecion of P 5, 6, 6 ono Q, 4,8 ha he orhogonal projecion of a vecor b ono a vecor a is defined and denoed b orhab b projab. Remember P Q 9 Proj P Q, 4, 8 Q Q 8 Orh P P Proj P 76 / 8, 60 / 8, 54 / 8 Q Q 5. a) Find he area of he parallelogram wih verices P(0, 0, 0), Q(-5,, ), R(-5, 0, ) and S(-0,, 4). PQ 5,,, PR 5, 0, PQ PR i j k 5 i 0 j 5k 5 0 Area= PQ PR = 6 sq. ni b) Find he area of he riangle wih verices P(5, 4, -5), Q(,, -6), and R(,, -8). PQ,,, PR,, PQ PR i j k 7i j-5k Area= PQ PR = 8 / sq. ni 6. For wha vales of are he vecors a 5,, and b 5,, orhogonal? For orhogonali a b = 5++ 0 has no real solion. Therefore is no sch vale we have.
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz 7. a) Find he disance of he poin P ( 5,,) from he line which passes hrogh he wo given poins Q (0,,) and R (,0, ). i j k PQ QR 5 0 i+0j+k PQ QR isance d= = 7/7 ni QR b) A man sared walking from a poin (-, ) and reached a he poin (-, 5). The man sared walking in he direcion of he vecor 5, a and changes his direcion onl once, when he rns a a righ angle. Wha are he coordinaes of he poin where he makes he rn? Rond or answer o wo decimal places. r( ) 5, and he disance d is given as d ( 5 ) (4 ) has minimm a 0.448 (check i) The reqired poin where he rns a righ angle is ( -0.76,.90) 8. A horizonal clohesline is ied beween poles, 8 meers apar. When a mass of kilograms is ied o he middle of he clohesline, i sags a disance of meers. Wha is he magnide of he ension on he ends of he clohesline? Solion: T T cos i T sin j = T ( 9 / 85i+/ 85j) W= 9.8=T T 45.8 85 T T 9. Find he parameric eqaions of he line hrogh he poin (0,,) ha is perpendiclar o he line,, z and parallel o he plane z 0.,,,,,, The eqaion of he line is L 0,,,,,, z kg
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz 0. Consider he pah r( ) (8, 9, 9ln ), 0. Find he lengh of he crve beween he poins P (7,44,9ln 4) and Q (8,9,0) Solion: Tes solion r ( ) 8,8,9 /, r ( ) 9( / ) 4 L 9( / ) d 9(5 ln 4) 47.48. A) Le f (, ) e. a) Find f (,) and b) f(,) in he direcion of v i j, where f (, ) e ( ), e, f (,) e, e, /, / f e / e v v B) Le f (, ) e. a) Find f (,) and b) f(,) in he direcion of v i j, where v v Answer: a) f (,) e, e b) f (,) e ( / ). A) Find he eqaion of he angen plane o z f (, ) e a he poin (,,). And approimae he vale of f (., 0.98) sing linear approimaion. f (, ), f (., 0.98) (.) ( 0.987).4 B) Find he eqaion of he angen plane o z f (, ) e a he poin (,,). And approimae he vale of f (0.97,.) sing linear approimaion. f e f (, ) f e f (, ) z f ( ) ( ) f (0.97,.) (0.97) (.) 0.54
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz. a) Sppose ha r z e cos, r s, s. Find (,) or answer o wo decimal places. z z, (,) rond s Solion: z r r ( s, ) se cos e ( sin ) s z (,).47 z r r s (,) e cos e ( sin ).47 s s b) Le W( s, ) F( ( s, ), v( s, )), where W, and v are differeniable, (, 0), s (, 0), (, 0) 6, v(, 0), vs (, 0) 5, v (, 0) 4. Frher given ha F (,), F (,) 0. Find W (,0), W (,0). v s Solion: 4. A) Given W W W v (,0) ( ) 0(5) 5 s s v s W W (, 0) (6) 0(4) 4 f (, ). Find he differenial dz in erms of and. dz d d d d ( ) d ( ) d B) Given f (, ). Find he differenial dz in erms of and. dz d d d d ( ) d ( ) d 5. The dimensions of a closed recanglar bo are measred as 90 cm, 50 cm, and 60 cm respecivel wih he error in each measremen a mos 0. cm. Use differenial o esimae he maimm error in calclaing he srface area of he bo. o no forge he ni. Solion: S lw lh wh ds ldw wdl ldh hdl wdh hdw (90 50 90 60 50 60)0. 60 sq. cm
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz 6. Verif if he fncion (, ) sin( a ) saisfies he wave eqaion a 0. a a a a cos( a), sin( a) a 0, es, verifies. cos( ), sin( ) 7. Find all he relaive maimm and minimm and saddle poins on he given 4 4 srface z f (, ) 4. o no forge o check he sign of he discriminan a each saionar poin. Wrie or answer in he bo below. Answer: f f 4 4 0, 4 4 0, f, f, f 4, f f [ f ] We have he solion (0, 0), (, ) and (-, -) as saionar poins Check ha saddle poin a (0, 0, ), min a (, ) and (-, -) and no ma. 8. Evalae he inegral ( ) da, where is he region bonded b he parabolas,. Yo ma rond or answer o wo decimal places. P or answer in he bo below Solion: ( ) da dd. 9. A) Evalae he inegral b reversing he order of he inegral Wrie or answer in he bo below. Solion: / e dd e dd e d 50.5 0 0 e dd. 0
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz B) 8 0 4 4 4 e dd e dd e d 0 57.8 0. A) Find 0 in he bo below. Solion: B) dd / 4cos 0 4 4 Tes solion: dd sing polar coordinaes. Wrie answer dd rdrd r r drd. 8/ 9 4. ( ) 4 Wrie down he doble inegral and also riple inegral o find he volme of he erahedron bonded b he coordinae planes and he plane z 4 4. Evalae boh he inegrals. E Complee he limi: zda zdd (4 4 ) dd 4 /. Complee he limi: E 4 4 dv dzdd dzdd E 0 4/.. Find he volme of he solid ha lies wihin he sphere he -plane, and below he cone z z 4, above
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Solion: raw he picre and idenif he region. / V dv d d d E / 4 8 sin.85. A sphere has cener a (0, 0, 0) and radis eqal o m. Use spherical coordinae o evalae he volme of he sphere Solion: he sphere has he eqaion z m m 4 m V sin d d d 4.9m 0 4. Evalae he line inegral sin d cos d, where C is he arc consising of he C op half of he circle from (, 0) o (-, 0) and he line segmen from (-, 0) o (-, ). Wrie eac answer, no decimals. Solion: Check ha on C crve: cos, sin, 0 and C C on C crve:,, 0 sin d cos d sin d cos d sin d cos d C C C sin(cos )( sin ) d cos(sin )(cos ) d sin( )( d) cos( )d cos cos sin 5. a) Show ha he line inegral ( e ) d e d is independen of pah, where C C is an pah from (0, ) o (, ). b) Use line inegral wih parameric represenaion for C o evalae inegral. Q P Solion: P e, Q e e C 0, he pah C is independen. ( e ) d e d ( ( ) e e ) d / e Where C has he parameric represenaion: r( ) 0, (,),
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz 6. Use Greens heorem o evalae he line inegral C 4 d d, where C is he rianglar pah consising of he line segmens from (0, 0) o (, 0), from (, 0) o (0, ), and from (0, ) o (0, 0). (0, ) 4 Q P Solion: d d da dd / 6 Where C Q P 4, Q P (0,0) (,0) 7. Show ha he vecor field (,, ) i ( z z F z e )j e k is conservaive. Also find he scalar fncion f (,, z ) sch ha grad f F. Solion: i j k Crl F F o z e e For scalar fncion we se he following resls; z z f f e, f e z z z he field is conservaive f h(, z) f h (, z) e, h(, z) e g( z ) Now z z f e g( z) f e g ( z) g( z) k z z z f (,, z) z e C Ths he scalar fncion is 8. Find he fl of he vecor field F(,, z) zi j k across he ni sphere cener a he origin. Solion: Use sin cos, sin sin, z cos,0,0 and r(, ) i j zk isin cos jsin sin k cos, r r isin cos j sin sin k sin cos F( r(, ) icos jsin sin k sin cos Now Check he inegral. S F(,, z) ds F n ds ( ) (sin cos cos sin sin ) F r r da d d 4 / 4.9
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Qiz : Febrar, 0. Name. Find he eqaion of he larges sphere cener a (, 4, 6) conained in he firs ocan. Solion: The sphere conained in he firs ocan ms have radis nis. The eqaion of he sphere is ( ) ( 4) ( z 6) 4. A man walks de eas on a deck of a ship a 5 miles per hor. The ship is moving norh a a speed of miles per hor. Find he speed and direcion (in radians) of he man relaive o he srface of he waer. Solion: 5,0, v 0, The reslan veloci is w v 5,, 5, The speed w 5 46 (0,) The direcion wih norh: Qiz : March 5, 0 cos 0.4 46 (5,0) Se A. Evalae dd d d 0 0 9ln 5. Evalae da, where {(, ) 0,,, z 0} da ( ) dd 5/ 6 =. Evalae 0 00 0 e dd b changing he order of inegraion. e dd e dd e 4. Evalae b sing polar coordinae: problem nmber 0(B) in es. dd 6, compare wih
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Se B. Evalae 0 0 dd d d 9ln 5. Evalae 0 0 e dd b changing he order of inegraion. e e dd compare wih se A.. Evalae da, where {(, ) 0,,, z 0} da ( ) dd 5/ 6 00 4. Evalae b sing polar coordinae: Qiz : April, 0 / cos dd r drd dd 6 / 9. Given ha r (,, z) i j z k, se spherical coordinae for a ni sphere cener a origin o find he normal vecor r r. Solion: r (,, z) i j zk r (, ) isin cos j sin sin k cos r (, ) i cos cos j cos sin k sin r (, ) i sin sin j sin cos r r i j k sin cos sin sin sin cos ( / / ), 0, 0. Evalae ds, where S is he srface z. S Rond or answer correc o hree decimal places. Solion: ds da dd 0.7 S
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz. Find he fl of he vecor field F(,, z) zi j k across he ni sphere cener a he origin. Use sin cos, sin sin, z cos,0,0 and r(, ) i j zk isin cos jsin sin k cos, r r isin cos j sin sin k sin cos F( r(, ) icos jsin sin k sin cos Now Check he inegral. Pop Qiz S F(,, z) ds F n ds 4 / ( ) (sin cos cos sin sin ) F r r da d d Answers o seleced homework problems: Secion:.5 0. 4. 4. 5 6. 7 8 8. 44 0. 8 Look a he domains: z z z z From diagram A: omain is plane A B C [ f (,, z) dz] dd [ f (,, z) dz] dd From diagram B: omain is z plane z [ f (,, z) d] dzd [ f (,, z) d] ddz From diagram C: I has wo separae regions for domain
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Secion:.6 z z [ f (,, z) d] ddz [ f (,, z) d] ddz z 0 z [ f (,, z) d] dzd [ f (,, z) d] dzd 0 8. A hperboloid of one shee wih ais he z ais.. A cone opening pward in he firs ocan. 6. 7 8. 0. 65 4. 4 (8 / ) 8. 6 5 5 Secion:.7 8. he srface of a sphere of radis cener a (0, 0, ) 0. a) b) sec co csc 4. 6. 8 6. 0 (5e ) Chaper Secion:. 4. Choose differen and vales and he lengh of he vecor, hen plo hem (look a eample in or e book) 4. cos sin sin 6. f ( ) ( ), plo orself z z z z Secion:. 6. cos cos cos 0. (4 / ) 6 4. 5/ 6. On C negaive, on C posiive 0. 4. (5 cos cos 4) Secion:.. 6 8. f (, ) ln K z 6. a) f (,, z) e ze b) e 8. /e
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Secion:.4. 0. 0 4. -6 6. 8. Secion:.5 4. a) (sin z cos ) cos z sin z b) 0 8. a) greaer han zero b) is zero 0. Tr orself 6. f (,, z) sin cos z K Secion:.6.Porion of ellipic clinder,0 z 6.,, z 4 4 9. scos, ssin, z scos, 0 s ; 0 / / 0. z 0 8. (6 0 ) 4 Secion:.7 6. 4 8. 6 6 4. (8 ) 4. 4. 0 408. Concep:. Evalaion of doble inegrals and riple inegrals. Clindrical and spherical coordinaes. Vecor fields, scalar fncions, line inegral and fndamenal heorem of line inegral, conservaive vecor field, 4. Greens heorem 5. Crl and divergence 6. Parameric srfaces and srface inegrals