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

1 Answer
Yonas Yohannes
Mar 22, 2016

#L ~~ 1.430#

Explanation:

Use the Arc Length theorem: Let f(x) be a continuous on [a, b], then the length of the curve y = f(x), a ≤ x ≤ b, is
#L =int_a^b sqrt(1 + [(df(x))/(dx)]^2)dx#
Now #f(x) = x^3-e^x; x in [-1,0] #
find #f'(x) =(df(x))/(dx)=3x^2-e^x #
#[(df(x))/dx]^2 = [3x^2-e^x]^2 #
#L=int_-1^0 sqrt(1+[3x^2-e^x]^2)dx# There is no closed form antiderivative so integrate using an integral calculator or estimate numerically:
#L ~~ 1.430#

In general the Arc Length integral is evaluated numerically, there very few Arc Length Integral with a closed form antiderivative.

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