What is the arclength of #f(x)=x^3-xe^x# on #x in [-1,0]#?

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Answer 1 · 2016-01-17T09:41:28

Arc length #s=1.54116# units

The formula to determine length of arc s:
#s=int_a^bsqrt(1+f' (x)^2) dx#
#a=-1# and #b=0# the limits

#f(x)=x^3-x e^x#

#f' (x) 3 x^2 -[x*e^x*1+1*e^x]#
#f' (x) = 3x^2 - x e^x - e^x#

#s=int_-1^0 sqrt(1+(3x^2 - x e^x - e^x)^2) dx#

There is no simple formula to evaluate the integral, so try using Simpson's Rule:

#s= 1.54116# units

Just observe the graph from #x=-1# to #x=0#
graph{y=x^3-x e^x [-2.5, 2.5, -1.25, 1.25]}