Arc Length And Curvature

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  1. Find the length of the curve

    rt=2sint5t2cost

    for 10t10 .

    2029
  2. Find the length of the curve

    rt=12ti+8t3/2j+3t2k

    for 0t1 .

    rt=12832t6t=1212t6t .

    So the arc length

    L=01rtdt=01122+122t+36t2dt=6012+tdt=15 .

  3. Find the length of the curve

    rt=t222t+13/23

    for 0t2 .

  4. Parametrize the curve

    rt=2ti+13tj+5+4tk

    with respect to arc length measured from the point where t=0 in the direction of increasing t .

    rts=229si+1329sj+5+429sk
  5. Parametrize the curve

    rt=e2tcos2ti+2j+e2tsin2tk

    with respect to arc length measured from the point where t=0 in the direction of increasing t .

  6. A particle starts out at the point 003 and moves 5 units along the the curve x=3sint,y=4t,z=3cost in the positive direction. Where is it now?
    3sin143cos1
  7. Compute the arc length of the curve defined by the vector-valued function

    rt=t2i2tj+lntk

    from the point 120 to the point e22e1 .

    e2