ESE 318-02, Spring 2016 Homework Set #11 Due Tuesday, Apr. 12 1. Zill 9.8.10. 2. Zill 9.8.11. 3. Zill 9.8.16. 4. Zill 9.8.18. 5. Zill 9.8.20. 6. Zill 9.8.22. 7. Zill 9.8.28. 8. Zill 9.8.30. 9. Zill 9.9.5. 10. Zill 9.9.8. 11. Zill 9.9.12. 12. Zill 9.9.13. 13. Zill 9.9.19. 14. Zill 9.9.24. 1. Zill 9.8.10. Evaluate the integral( )∫++Cxydydxyx2, over the curve C as pictured. ( )[ ] [ ] [ ]( )302514222202:300:201:150221220221502102=+−+=++=++−=++=====−=−−∫∫∫∫yxyydyxdxydyxydydxyxdxxLegdyyLegdxxLegCESE 318-02, Spring 2016 2. Zill 9.8.11. ( )1321021031021022===+=+≤≤=⇒=∫∫∫xdxxxdxxdxxxdyydxxxdxdyxyC Also, F = y, x r = x, x2F = x2, x dr = 1, 2x dxF ⋅ drC∫= x2+ 2x2( )dx01∫= 3x2dx01∫= x301=1 3. Zill 9.8.16 Evaluate ∫+−Cxydydxy2, where C is given by 2023≤≤== ttytx. ( )( )( )( )[ ]( )75127446232232207206206620232322===+−=+−=+−==∫∫∫∫tdttdtttdttttdttxydydxydttdydtdxC Alternatively, ( )7512744623,22,3,22,,2,20720620662024624632===+−=⋅−=⋅=ʹ−=⇒=−=∫∫∫∫tdttdtttdttttdrFtrttFttrxyyFC 4. Zill 9.8.18. Evaluate ∫+Cydyxdx 24 C: ( ) ( )2,91,013tofromyx −+= 4x dx + 2y dyC∫= 4 y3+1( )3y2dy( )+ 2ydy⎡⎣⎤⎦−12∫= 12y5+12y2+ 2y( )dy−12∫= 2 y6+ 4 y3+ y2⎡⎣⎤⎦−12= 128 + 32 + 4( )− 2 − 4 +1( )=164 +1 =165 Alternatively, F = 4x, 2y r = y3+1, y ⇒ F = 4 y3+1( ), 2y dr = 3y2,1 dyF ⋅ drC∫= 4 y3+1( ), 2y ⋅ 3y2,1 dy−2∫= 12y5+12y2+ 2y( )dy02∫= 2y6+ 4y3+ y2⎡⎣⎤⎦−12= 128 + 32 + 4( )− 2 − 4 +1( )=164 +1 =165ESE 318-02, Spring 2016 5. Zill 9.8.20 Evaluate( )∫−+Cxydydxyx 222, C as pictured. ( ) ( ) ( ) ( )( )( )[ ]( ) ( ) ( )( )[ ]53:222222:3222:315331311066110501501224222533110553331104210242222−=−−−−=−==−+=−+=−=−=−=−+=−+=∫∫∫∫∫∫∫Tot alydyydyydyyyydyyyxydydxyxyxTopxxdxxxxdxxxdxxxxydydxyxxyBot tomCC Alternate: parameterize each leg in terms of x. (Bottom leg, same as before.) ( ) ( ) ( )( )( )5333:22:10410242102012012/1212/1222−=−=−−−==−+=−+∫∫∫∫∫∫−dxxdxxxxTotaldxxdxxdxxxxdxxxxydydxyxTopC Alternate: F = x2+ y2, −2xyBottom y = x2( ): r = x, x2⇒ F = x2+ x4, −2x3dr = 1, 2x dxF ⋅ drC∫= x2+ x4− 4x4( )dx01∫= x2− 3x4( )dx01∫=13x3−35x5⎡⎣⎤⎦01=13−35Top x = y2( ): r = y2, y ⇒ F = y4+ y2, −2y3dr = 2y,1 dyF ⋅ drC∫= 2y5+ 2y3− 2y3( )dy10∫= − 2y5dy01∫= −216y6⎡⎣⎤⎦01= −13Total :13−35−13= −35ESE 318-02, Spring 2016 6. Zill 9.8.22. Evaluate ∫−Cdyxydxyx232 on the closed curve shown. 0000:0404:0435120233640232===−==−==∫∫dydxxLeftxdxxdyyTop For the slanted leg, we have multiple parameterization options: y in terms of x, x in terms of y, or both in terms of some t. Here are the 1st two ways (not done in class). ( ) ( ) ( )[ ][ ]( ) ( ) ( )[ ][ ]335235123160316032564048164814032158140221213221213160325620448668203520232:32:3288222:2−=−=−=−=−=−==−=−=−=−=∫∫∫∫Tot alyydyyydyyyyyyxxxdxxxdxxxxxxy Also: r = 2ti + 4tj, 0 < t < 1. ( ) ( ) ( )( )[ ]3352316035123160325610441286651210354,20,03535352322320:321285121285124,232,25632,25642,42,4,24,2−=++−=−=−=−=⋅−=⋅−=⋅−=−=−===∫∫Tot alttdtttdrFttttdrFttttttxyyxFdrttr 7. Zill 9.8.28. I’ll parameterize each leg differently. 54302464860,8,66,0,80,8,6:0,8,6,,5010501021=+=+=+⋅=⋅======∫∫∫∫∫dztdtdzdtttdrFdydxyxrttrxzyFCESE 318-02, Spring 2016 8. Zill 9.8.30 Evaluate the work integral with F and r as indicated. 1032≤≤++=++= tktjttirkxyejxeieFxyzxyx ( )( )( ) ( )11163323,2,1,,3,2,1,,,,61321321021103210105631023210105210232323326363636363−=−++=++=++=++=⋅=⋅=⇒==⇒===∫∫∫∫∫∫eeeeedtetdtetdtedtetetedtttetteedrFdtttdrtttretteeFtztytxttttttttttttCttt 9. Zill 9.9.5 ( )341441410431444,1:,,14141224,41,44,41,422=+−=⎟⎟⎠⎞⎜⎜⎝⎛−==+−≤≤===+−=⋅=−=∇−=⇒−=∫∫∫ydyydyyxdxyydxxdrFFyxyCheckyxyxyxyFφφ 10. Zill 9.9.8 ( )( ) ( )( )( )( )( )( )27230142230130130,01,127250,01,134225322838445012401,10,084,45:24,84,45=+=−−=−−=−++−=⋅≤≤−−=−==−−−=−−=⋅−+=∇−+=⇒−+=−−−−−∫∫∫∫xxdxxxdxxxdxxxdrFxdxdyxydrFyxyxCheckyxyxyxyxyxFφφφφESE 318-02, Spring 2016 11. Zill 9.9.12 ( )( ) ( )( ) ( )checkxyxyxyxydyxyyxdxxyveconservatixyxQxyyPQjPijxyixyF13,2111326613222323232232322223+=∇+=+=+=∴=∂∂=∂∂+=++=∫∫φφ 12. Zill 9.9.13 ( )( )( )veconservatinotxQyPxyxyxyyxyyyxyxyxQxyyxyxyxyyxyxyyyPQjPijxyxyixyyF∴∂∂≠∂∂−−=−−=∂∂+−=+−=∂∂+=−=232222223222222cos2sin2sin2cos2cos2sin2cos22sinsin2cos 13. Zill 9.9.19 6318420,,,,,,8,4,21.1.1=−⋅⋅=⋅=−+−−=∂∂∂∂∂∂==∇=⇒=∫drFzzyyxxxyxzyzzyxkjialsoFxyxzyzxyzxyxzyzFφφESE 318-02, Spring 2016 14. Zill 9.9.24 Complete solution, from checking the curl through finding φ two ways. ( )( )( )( )( )( )( )( )( )( )( ) ( )[ ]( )( )(