Determining the Length of a Parametric Curve (Parametric Form)

Key Questions

• How do you find the length of the curve #x=3t-t^3#, #y=3t^2#, where #0<=t<=sqrt(3)# ?

  Answer:

  #6sqrt3#.

  Explanation:

  The answer is #6sqrt3#.

  The arclength of a parametric curve can be found using the formula: #L=int_(t_i)^(t_f)sqrt(((dx)/(dt))^2+((dy)/(dt))^2)dt#. Since #x# and #y# are perpendicular, it's not difficult to see why this computes the arclength.

  It isn't very different from the arclength of a regular function: #L=int_a^b sqrt(1+((dy)/(dx))^2)dx#. If you need the derivation of the parametric formula, please ask it as a separate question.

  We find the 2 derivatives:
  #(dx)/(dt)=3-3t^2#
  #(dy)/(dt)=6t#

  And we substitute these into the integral:
  #L=int_0^(sqrt3)sqrt((3-3t^2)^2+(6t)^2)dt#

  And solve:
  #=int_0^(sqrt3)sqrt(9-18t^2+9t^4+36t^2)dt#
  #=int_0^(sqrt3)sqrt(9+18t^2+9t^4)dt#
  #=int_0^(sqrt3)sqrt((3+3t^2)^2)dt#
  #=int_0^(sqrt3)(3+3t^2)dt#
  #=3t+t^3|_0^(sqrt3)#
  #=3sqrt3+3sqrt3#
  #=6sqrt3#

  Be aware that arclength usually has a difficult function to integrate. Most integrable functions look like the above where a binomial is squared and adding the two terms will flip the sign of the binomial.

  Amory W. · 3 · Aug 15 2014
• How do you find the length of the curve #x=1+3t^2#, #y=4+2t^3#, where #0<=t<=1# ?

  By taking the derivative with respect to #t#,

  #{(x'(t)=6t),(y'(t)=6t^2):}#

  Let us now find the length #L# of the curve.

  #L=int_0^1 sqrt{[x'(t)]^2+[y'(t)]^2}dt#

  #=int_0^1 sqrt{6^2t^2+6^2t^4} dt#

  by pulling #6t# out of the square-root,

  #=int_0^1 6t sqrt{1+t^2} dt#

  by rewriting a bit further,

  #=3int_0^1 2t(1+t^2)^{1/2}dt#

  by General Power Rule,

  #=3[2/3(1+t^2)^{3/2}]_0^1=2(2^{3/2}-1)#

  I hope that this was helpful.

  Wataru · 3 · Oct 10 2014
• How do you find the arc length of a parametric curve?

  Please see how I solved this problem, it has a full explanation.

  How do you find the length of the curve #x=3t-t^3#, #y=3t^2#, where #0<=t<=sqrt(3)# ?

  Amory W. · · Aug 17 2014

• How do you find the arc length of a parametric curve?
• How do you find the length of the curve #x=1+3t^2#, #y=4+2t^3#, where #0<=t<=1# ?
• How do you find the length of the curve #x=e^t+e^-t#, #y=5-2t#, where #0<=t<=3# ?
• How do you find the length of the curve #x=t/(1+t)#, #y=ln(1+t)#, where #0<=t<=2# ?
• How do you find the length of the curve #x=3t-t^3#, #y=3t^2#, where #0<=t<=sqrt(3)# ?
• How do you determine the length of a parametric curve?
• How do you determine the length of #x=3t^2#, #y=t^3+4t# for t is between [0,2]?
• How do you determine the length of #x=2t^2#, #y=t^3+3t# for t is between [0,2]?
• What is the arc length of #r(t)=(t,t,t)# on #tin [1,2]#?
• What is the arc length of #r(t)=(te^(t^2),t^2e^t,1/t)# on #tin [1,ln2]#?
• What is the arc length of #r(t)=(t^2,2t,4-t)# on #tin [0,5]#?
• What is the arc length of the curve given by #r(t)= (1,t,t^2)# on # t in [0, 1]#?
• What is the arc length of the curve given by #r(t)= (1/t,1,t)# on # t in [0, 1]#?
• What is the arc length of the curve given by #r(t)= (ln(1/t),t^2,t)# on # t in [1, 10]#?
• What is the arc length of the curve given by #r(t)= (e^-t,e^t,1)# on # t in [1, 2]#?
• What is the arc length of the curve given by #r(t)= (-t,2t^2,-t^3)# on # t in [3,4]#?
• What is the arc length of the curve given by #r(t)= (9sqrt(2),e^(9t),e^(-9t))# on # t in [3,4]#?
• What is the arc length of the curve given by #r(t)= (e^tsqrt(2t),e^t,e^(-2t))# on # t in [3,4]#?
• What is the arclength of #(sint/(t+cos2t),cost/(2t))# on #t in [pi/12,pi/3]#?
• What is the arclength of #(tant-sect*csct)# on #t in [pi/8,pi/3]#?
• What is the arclength of #(tant,sect*csct)# on #t in [pi/8,pi/3]#?
• What is the arclength of #(t-e^(2t),lnt)# on #t in [1,12]#?
• What is the arclength of #(t^2-t,t^2-1)# on #t in [-1,1]#?
• What is the arclength of #(t,t-1)# on #t in [-7,1]#?
• What is the arclength of #(3t^2,t^4-t)# on #t in [-4,1]#?
• What is the arclength of #(2t^2-t,t^4-t)# on #t in [-4,1]#?
• What is the arclength of #(2t^3-t^2+1,2t^2-t+4)# on #t in [-2,1]#?
• What is the arclength of #(9t^3-2t^2+15,-2t^2-12t-1)# on #t in [-2,3]#?
• What is the arclength of #(sqrt(t+2),1/t)# on #t in [1,3]#?
• What is the arclength of #(sqrt(3t-2),1/sqrt(t+3))# on #t in [1,3]#?
• What is the arclength of #(t/(5-t),8/t)# on #t in [1,4]#?
• What is the arclength of #(t/(1+t)^2,-1/t)# on #t in [2,4]#?
• What is the arclength of #(e^(2t)-t,t-t/e^(t-1))# on #t in [-1,1]#?
• What is the arclength of #(e^(-2t)-t^2,t-t/e^(t-1))# on #t in [-1,1]#?
• What is the arclength of #(t/(t+5),t)# on #t in [-1,1]#?
• What is the arclength of #(t-t^3,t+1)# on #t in [-2,4]#?
• What is the arclength of #(t-2t^3,4t+1)# on #t in [-2,4]#?
• What is the arclength of #(sqrtt,1/sqrt(t^2+3))# on #t in [1,2]#?
• What is the arclength of #(1/t^2,1/t)# on #t in [1,2]#?
• What is the arclength of #(1/(1-e^t),2t)# on #t in [2,4]#?
• What is the arclength of #(1/(t+te^t),-t)# on #t in [3,4]#?
• What is the arclength of #(t-3,t^2)# on #t in [1,2]#?
• What is the arclength of #(t-3t^2,t^2-t)# on #t in [1,2]#?
• What is the arclength of #(lnt/t,lnt)# on #t in [1,2]#?
• What is the arclength of #(t^2lnt,t-lnt)# on #t in [1,2]#?
• What is the arclength of #(t^2lnt,(lnt)^2)# on #t in [1,2]#?
• What is the arclength of #(t^2lnt,lnt^2)# on #t in [1,2]#?
• What is the arclength of #(t^2-lnt,lnt)# on #t in [1,2]#?
• What is the arclength of #(ln(t+3),lnt/t)# on #t in [1,2]#?
• What is the arclength of #((e^tlnt)/t,t/(e^tlnt))# on #t in [2,4]#?
• What is the arclength of #(t-3,t+4)# on #t in [2,4]#?
• What is the arclength of #(t^2-t-1,9t^2-2/t)# on #t in [1,4]#?
• What is the arclength of #(t-1,t/(t+5))# on #t in [-1,1]#?
• What is the arclength of #(sqrt(e^(t^2)/(e^t)+1),t^3)# on #t in [-1,1]#?
• What is the arclength of #f(t) = (t-e^(t),t-2/t+3)# on #t in [1,2]#?
• What is the arclength of #f(t) = (t,t,t)# on #t in [1,3]#?
• What is the arclength of #f(t) = (t^3-t^2+5t,9t)# on #t in [1,4]#?
• What is the arclength of #f(t) = (-2(t+3)^3,(t-2)^2)# on #t in [1,2]#?
• What is the arclength of #f(t) = (-(t+3)^2,3t-4)# on #t in [0,1]#?
• What is the arclength of #f(t) = (t-2,1/(t-2))# on #t in [0,1]#?
• What is the arclength of #f(t) = (tlnt,t/lnt)# on #t in [1,e]#?
• What is the arclength of #f(t) = (t+sqrt(lnt),t-sqrtlnt))# on #t in [1,e]#?
• What is the arclength of #f(t) = (tsqrt(lnt),t^3/(4-t))# on #t in [1,e]#?
• What is the arclength of #f(t) = (lnt/t,ln(t+2))# on #t in [1,e]#?
• What is the arclength of #f(t) = (sqrt(t-2),t^2)# on #t in [2,3]#?
• What is the arclength of #f(t) = (t^2sqrt(t-1),t^2+t-1)# on #t in [2,3]#?
• What is the arclength of #f(t) = (t-sqrt(t-1),t^2/(t^2-1))# on #t in [2,3]#?
• What is the arclength of #f(t) = (t-tsqrt(t-1),t/(t^2-1))# on #t in [2,3]#?
• What is the arclength of #f(t) = (t/sqrt(t-1),t/(t^2-1))# on #t in [2,3]#?
• What is the arclength of #f(t) = (t^3-1,t^2-1)# on #t in [2,3]#?
• What is the arclength of #f(t) = (t^3-t+55,t^2-1)# on #t in [2,3]#?
• What is the arclength of #f(t) = (te^(2t)-e^t-3t,-2t^2)# on #t in [1,3]#?
• What is the arclength of #f(t) = (e^(t)-e^(t^2)/t,t^2-1)# on #t in [1,3]#?
• What is the arclength of #f(t) = (t-7,t+7)# on #t in [1,3]#?
• What is the arclength of #f(t) = (t^2-5,e^t)# on #t in [1,3]#?
• What is the arclength of #f(t) = (t^3-t^2+t,t-t^2)# on #t in [0,1]#?
• What is the arclength of #f(t) = (sqrt(t^2-2t+1),t^2-2t+1)# on #t in [0,1]#?
• What is the arclength of #f(t) = (sqrt(t^2-t^3),t^3-t^2)# on #t in [-1,1]#?
• What is the arclength of #f(t) = (t-sqrt(t^2+2),t+te^(t-2))# on #t in [-1,1]#?
• What is the arclength of #f(t) = (t/sqrt(t^2+2),t/e^(t-2))# on #t in [-1,1]#?
• What is the arclength of #f(t) = (sint*cos2t,sint)# on #t in [-pi,0]#?
• What is the arclength of #f(t) = (sin^2t-cos2t,t/pi)# on #t in [-pi/4,pi/4]#?
• What is the arclength of #f(t) = (sin^2t/sin(2t),tant-sec^2t)# on #t in [pi/12,pi/4]#?
• What is the arclength of #f(t) = (cos2t-sin2t,tan^2t)# on #t in [pi/12,(5pi)/12]#?
• What is the arclength of #f(t) = (sin2t-tcsct,t^2-1)# on #t in [pi/12,(5pi)/12]#?
• What is the arc length of #f(t)=(3t-4,t^3-2t) # over #t in [-1,2]#?
• What is the arc length of #f(t)=(t^2-4t,5-1/t) # over #t in [3,4] #?
• What is the arc length of #f(t)=(t^2-4t,5-t) # over #t in [3,4] #?
• What is the arc length of #f(t)=(lnt,5-lnt) # over #t in [3,4] #?
• What is the arc length of #f(t)=(ln(1/t) ,5-lnt) # over #t in [3,4] #?
• What is the arc length of #f(t)=((4-t^2)e^t,te^t) # over #t in [3,4] #?
• What is the arc length of #f(t)=(3te^t,t-e^t) # over #t in [2,4]#?
• What is the arc length of #f(t)=(sqrt(t-1),2-8t) # over #t in [1,3]#?
• What is the arc length of #f(t)=(1/sqrt(t+t^2),2-8t) # over #t in [1,3]#?
• What is the arc length of #f(t)=(t-tsqrt(t-1),8-2t) # over #t in [1,4]#?
• Consider the parametric equation #x= 10(cost+tsint)# and #y= 10(sint-tcost)#, What is the length of the curve from #0# to #((3pi)/2)#?
• Question #89633
• Find the exact length of the curve (parametrics)?
• Parametric arc length?
• Find arc length given #x=t\sint#, #y=t\cost# and #0\let\le1#?