Question

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

Answer 1

`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.