What is the arclength of #(sqrt(e^(t^2)/(e^t)+1),t^3)# on #t in [-1,1]#?

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Answer 1 · 2016-01-15T09:54:22

#s=2.86658# units

Given #x(t)=sqrt(e^(t^2-t) +1)# and # y(t)=t^3#

#dx/dt=((e^(t^2-t))(2t-1))/(2 sqrt(e^(t^2-t) +1)#

#dy/dt = 3 t^2#

#s = int_-1^1 sqrt((dx/dt)^2 + (dy/dt)^2) dt#

#s = int_-1^1 sqrt( (( ( e^(t^2-t) )(2t-1))/(2 sqrt(e^(t^2-t) +1)))^2 + (3 t^2)^2) dt#

the integral is complicated so Simpson's Rule is used:

so that #s=2.86658#