Evaluate the line integral â«CFâ dr, where F(x,y,z)=â4sinxi+cosyjâxzk and C is given by the vector function r(t)=t6iât5j+t4k , 0â¤tâ¤1.
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Evaluate the line integral â«CFâ d râ«CFâ d r where F=â©4sinx,cosy,xzâªF=â©4sinâ¡x,cosâ¡y,xz⪠and CC is the path given by r(t)=(3t3,â2t2,â2t)r(t)=(3t3,â2t2,â2t) for 0â¤tâ¤1
Evaluate the line integral â«CFâ d r where F=â©2sinâ¡x,â3cosâ¡y,5xz⪠and C is the path given by r(t)=(â2t^3,â2t^2,3t) for 0â¤tâ¤1
F(x,y)=(4xy3+8)i+(6x2y2+2e2y)jF(x,y)=(4xy3+8)i+(6x2y2+2e2y)j
is conservative by finding a potential function ff for FF, and use ff to compute â«CFâ drâ«CFâ dr, where CC is the curve given by
r(t)=2sin9ti+2tÏsin105tjr(t)=2sin9â¡ti+2tÏsin10â¡5tj
for 0â¤tâ¤Ï/20â¤tâ¤Ï/2.