Question Solved1 Answer 15. If F(r) = (2y+3)i +xzj+(yz – x)k, evaluate the line integral along the following paths C: (a) x = 2t², y = t, z = t3 from t = 0 to t = 1. (b) The straight line joining (0,0,0) to (2,1,1). (c) The straight lines from (0,0,0) to (0,0,1), then to (0,1,1) and then to (2,1,1).

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Transcribed Image Text: 15. If F(r) = (2y+3)i +xzj+(yz – x)k, evaluate the line integral along the following paths C: (a) x = 2t², y = t, z = t3 from t = 0 to t = 1. (b) The straight line joining (0,0,0) to (2,1,1). (c) The straight lines from (0,0,0) to (0,0,1), then to (0,1,1) and then to (2,1,1).
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Transcribed Image Text: 15. If F(r) = (2y+3)i +xzj+(yz – x)k, evaluate the line integral along the following paths C: (a) x = 2t², y = t, z = t3 from t = 0 to t = 1. (b) The straight line joining (0,0,0) to (2,1,1). (c) The straight lines from (0,0,0) to (0,0,1), then to (0,1,1) and then to (2,1,1).
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Solution: F(x)=(2y+3) hat(ı)+xz^(˙) hat(ȷ)+(yz-x) hat(k)(a) T0. find line integral along the path C x=2t^(2),y=t,quadz_(0)=t^(3)dots (1) from t=0 to t=1" Along "{:[c","F=(2y+3) hat(ı)+xz hat(ȷ)+(yz-x) hat(k)],[=(2t+3) hat(ı)+2t^(5) hat(ȷ)+(t^(4)-2t^(2)) hat(k)dots" From(1) "],[" and "r=x hat(ı)+y hat(ȷ)+z hat(k).]:}{:[r=2t^(2) hat(ı)+t hat(ȷ)+t^(3) hat(k)dotsquad from(1)],[" i.c "dr=(4t( hat(ı))+( hat(ȷ))+3t^(2)( hat(k)))dt]:}Then, quadint_(c)F*dr=int_(0)^(1)(2t+3)4tdt+2t^(5)dt+(t^(4)-2t^(2))3t^(2)dt{:[=int_(0)^(1)(8t^(2)+12 t)dt+2t^(5)dt+(3t^(6)-6t^(4))dt],[=[(8t^(3))/(3)+(12t^(2))/(2)+(2t^(6))/(6)+(3t^(7))/(7)-(6t^(5))/(5)]_(0)^(1)],[=(8)/(3)+(12)/(2)+(2)/(3)+(3)/(7)-(6)/(5)],[=(899)/(105)=8.56]:}(b) The straight line joining (0,0,0) to (2,1,1) The line integral along the above line is found by convortang it into its parametric form, i.c{:[x=2t","y=t","z=t],[" Here "quadr=x hat(ı)+y hat(ȷ)+z hat(k)],[r=2t hat(ı)+t hat(ȷ)+t hat(k)],[dr=(2 hat(ı)+ hat ... See the full answer