Line Integrals

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  1. Evaluate the line integral Cyds , where C is the curve x=t2 , y=t , 0t2 .
  2. Evaluate the line integral Cxy4ds , where C is the right half of the circle x2+y2=16 .
  3. Evaluate the line integral Cxy3ds , where C is the curve x=4sint , y=4cost , z=3t , 0tπ/2 .
  4. Evaluate the line integral Cxeyzds , where C is the line segment from 000 to 123 .
  5. Evaluate the line integral Cxydx+xydy where C consists of line segments from 00 to 20 and from 20 to 32 .
  6. Evaluate the line integral Csinxdx+cosydy where C consists of the top half of the unit circle from 10 to 10 and the line segment from 10 to 23 .
  7. Evaluate the line integral Cx2dx+y2dy+z2dz , where C consists of line segments from 000 to 121 , and from 121 to 320 .
  8. Evaluate the line integral CFdr where Fxy=x2y3iyxj and C is given by the vector function rt=t2it3j , 0t1 .
  9. Evaluate the line integral CFdr where Fxyz=zyx and C is given by the vector function rt=tsintcost , 0tπ .
  10. A thin wire is bent into the shape of a semicircle x2+y2=4 , x0 . If the linear density is a constant k , find the mass and the centroid of the wire.
    2πk and 4/π0
  11. Find the work done by the force field Fxyz=y+zx+zx+y on a particle that moves along the line segment from 100 to 342 .
    1. An intelligent robot ( 160 kg) is carrying a canister with oil ( 25 kg) up a helical ramp that encircles a rocket with a radius 20 m. If the rocket is 90 m high and the robot makes exactly 3 complete revolutions, how much work is done by the robot against gravity while climbing to the top?
    2. If there is a hole in the canister and 9 kg of oil leaks out at a steady rate during the ascent, how much work is done?
  12. An object moves along the curve C shown in the figure from 12 to 98 . The lengths of the vectors in the force field F are measured in newtons by the scales on the axes. Estimate the work done by F on the object.

    discrete force field

    22 J