# Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-18 JWDD027-Salas-v1 January 4, 2007 18:12 918 SECTION 18.1 CHAPTER 18 SECTION 18.1 1. (a) h ( x,y )= y i + x j ; r ( u u i + u 2 j ,u [0 , 1] x ( u u, y ( u u 2 ; x 0 ( u )=1 ,y 0 ( u )=2 u h ( r ( u )) · r 0 ( u y ( u ) x 0 ( u )+ x ( u ) y 0 ( u u 2 (1) + u (2 u )=3 u 2 Z C h ( r ) · dr = Z 1 0 3 u 2 du =1 (b) h ( y i + x j ; r ( u u 3 i 2 u j , x ( u u 3 ( u 2 u ; x 0 ( u u 2 0 ( u 2 h ( r ( u )) · r 0 ( u y ( u ) x 0 ( u x ( u ) y 0 ( u )=( 2 u )(3 u 2 u 3 ( 2) = 8 u 3 Z C h ( r ) · dr = Z 1 0 8 u 3 du = 2 2. (a) Z C h · dr = Z 1 0 ( u i + u 2 j ) · ( i +2 u j ) du = Z 1 0 ( u u 3 ) du (b) Z C h · dr = Z 1 0 ( u 3 i 2 u j ) · (3 u 2 i 2 j ) du = Z 1 0 (3 u 5 +4 u ) du = 5 2 3. h ( y i + x j ; r ( u ) = cos u i sin u j , 2 π ] x ( u ) = cos u, y ( u sin u ; x 0 ( u sin u, y 0 ( u cos u h ( r ( u )) · r 0 ( u y ( u ) x 0 ( u x ( u ) y 0 ( u ) = sin 2 u cos 2 u Z C h ( r ) · dr = Z 2 π 0 (sin 2 u cos 2 u ) du =0 4. (a) Z C h · dr = Z 1 0 ( e u i j ) · ( e u i e u j ) du = Z 1 0 (1 2 e u ) du =2 e 1 1 (b) Z C h · dr = Z 2 0 2 j · (1 u ) i du = Z 2 0 0 du 5. (a) r ( u )=(2 u ) i +(3 u ) j , Z C h ( r ) · dr = Z 1 0 ( 5+5 u u 2 ) du = 17 6 (b) r ( u ) = (1 + u ) i +(2+ u ) j , Z C h ( r ) · dr = Z 1 0 (1+3 u + u 2 ) du = 17 6

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-18 JWDD027-Salas-v1 January 4, 2007 18:12 SECTION 18.1 919 6. (a) Z C h · dr = Z 4 1 µ 1 u (1 + u ) i + 1 u 1+ u j · µ 1 2 u i + 1 2 u j du = Z 4 1 1 u (1 + u ) du =ln 8 5 (b) Z C h · dr = Z 1 0 1 (1 + u ) 3 ( i + j ) · ( i + j ) du = Z 1 0 2 (1 + u ) 3 du = 3 4 7. C = C 1 C 2 C 3 where, C 1 : r ( u )=(1 u )( 2 i )+ u (2 i )=(4 u 2) i ,u [0 , 1] C 2 : r ( u u )(2 i u (2 j )=(2 2 u ) i +2 u j , C 3 : r ( u u )(2 j u ( 2 i )= 2 u i +(2 2 u ) j , Z C = Z C 1 + Z C 2 + Z C 3 =0+( 4)+( 4) = 8 8. r ( u )=( 1+2 u ) i +(1+ u ) j [0 , 1] Z C h · dr = Z 1 0 ( e 2+ u i + e 3 u j ) · (2 i + j ) du = Z 1 0 (2 e 2+ u + e 3 u ) du = e 5 e 2 +6 e 6 3 e 2 9. C 1 : r ( u u ) i , C 2 : r ( u ) = cos u i + sin u j ] Z C = Z C 1 + Z C 2 π π 10. Bottom: r ( u u i ; Z 1 0 u 3 j · i du = Z 1 0 0 du =0 Right side: r ( u i + u j ; Z 1 0 [3 u i +(1+2 u ) j ] · j du = Z 1 0 (1+2 u ) du =2 Top: r ( u u ) i + j ; Z 1 0 3(1 u ) 2 i · ( i ) du = Z 1 0 3(1 u ) 2 du = 1 Left: r ( u u ) j ; Z 1 0 2(1 u ) j · ( j ) du = Z 1 0 2(1 u ) du = 1 Z C h · d r = sum of the above = 0
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-18 JWDD027-Salas-v1 January 4, 2007 18:12 920 SECTION 18.1 11. (a) r ( u )= u i + u j + u k ,u [0 , 1] Z C h ( r ) · dr = Z 1 0 3 u 2 du =1 (b) Z C h ( r ) · dr = Z 1 0 (2 u 3 + u 5 +3 u 6 ) du = 23 21 12. (a) Z C h · dr = Z 1 0 e u ( i + j + k ) · ( i + j + k ) du = Z 1 0 3 e u du =3( e 1) (b) Z C h · dr = Z 1 0 ( e u i + e u 2 j + e u 3 k ) · ( i +2 u j u 2 k ) du = Z 1 0 ( e u ue u 2 u 2 e u 3 ) du e 1) . 13. (a) r ( u )=2 u i u j u k , Z C h ( r ) · dr = Z 1 0 (2 cos2 u + 3 sin3 u u 2 ) du = £ sin2 u cos3 u + u 3 ¤ 1 0 = 2 + sin2 (b) Z C h ( r ) · dr = Z 1 0 ( 2 u cos u 2 u 2 sin u 3 u 4 ) du = · sin u 2 cos u 3 1 5 u 5 ± 1 0 = 4 5 + sin1 cos1 14. (a) Z C h · dr = Z 1 0 ( 2 u 2 i +4 u 3 j 2 u 3 k ) · (2 i j + k ) du = Z 1 0 ( 4 u 2 6 u 3 ) du = 17 6 (b) Z C h · dr = Z 1 0 ( i + ue 2 u j + u k ) · ( e u i e u j + k ) du = Z 1 0 ( e u ue u + u ) du = e 3 2 15. r ( u u i + u 2 j , 2] Z C F ( r ) · dr = Z 2 0 £ ( u u 2 )+(2 u + u 2 )2 u ¤ du = Z 2 0 ( 2 u 3 +6 u 2 + u ) du =26 16. C 1 : r ( u u i ; Z 1 0 u i · i du = Z 1 0 udu = 1 2 C 2 : r ( u i + u j ; Z 1 0 (cos u i u sin1 j ) · j du = Z 1 0 u du =

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ch18[1] - P1 PBU/OVY JWDD027-18 P2 PBU/OVY JWDD027-Salas-v1...

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