# Determining the Length of a Parametric Curve (Parametric Form)

Determining the length of a parametric curve, expressed in its parametric form, is a fundamental task in mathematics with wide-ranging applications in fields such as physics, engineering, and computer graphics. Parametric equations represent curves as sets of equations, where each variable is expressed in terms of a common parameter. Calculating the length of such curves involves integrating the square root of the sum of the squares of the derivatives of the parameterized functions with respect to the parameter. This process enables precise measurement and analysis of complex curves, aiding in problem-solving and modeling endeavors.

Questions

- 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)#?
- What is the arclength of #f(t) = (sqrt(t-2),t^2)# on #t in [2,3]#?
- What is the arclength of #(t-3,t^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 #f(t) = (sqrt(t^2-2t+1),t^2-2t+1)# on #t in [0,1]#?
- 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 #(tant-sect*csct)# on #t in [pi/8,pi/3]#?
- What is the arc length of #f(t)=(sqrt(t-1),2-8t) # over #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) = (t-sqrt(t-1),t^2/(t^2-1))# on #t in [2,3]#?
- What is the arclength of #(t-e^(2t),lnt)# on #t in [1,12]#?
- What is the arclength of #(e^(2t)-t,t-t/e^(t-1))# on #t in [-1,1]#?
- What is the arclength of #f(t) = (t^3-1,t^2-1)# on #t in [2,3]#?
- What is the arc length of #r(t)=(t^2,2t,4-t)# on #tin [0,5]#?
- What is the arclength of #(t/(1+t)^2,-1/t)# on #t in [2,4]#?
- What is the arclength of #(sint/(t+cos2t),cost/(2t))# on #t in [pi/12,pi/3]#?
- What is the arclength of #(e^(-2t)-t^2,t-t/e^(t-1))# on #t in [-1,1]#?
- What is the arclength of #f(t) = (-(t+3)^2,3t-4)# on #t in [0,1]#?
- 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-sqrt(t^2+2),t+te^(t-2))# on #t in [-1,1]#?