Question

What is the arc length of `f(x)= sqrt(x^3+5)` on `x in [0,2]`?

Answer 1

The length is around `2.58`.

Explanation:

The formula for arc length of a curve of a function `f(x)` on the interval `[a,b]` is equal to:
`int_a^bsqrt(1+(f'(x))^2)\ dx`

Let us first work out `f'(x)`:
`d/dx(sqrt(x^3+5))=`

I will use the chain rule and let `u=x^3+5`
`=d/(du)(sqrtu)*d/dx(x^3+5)=1/(2sqrtu)*3x^2=`

`=(3x^2)/(2sqrt(x^3+5))`

Now we plug this into our arc length formula:
`int_0^2sqrt(1+((3x^2)/(2sqrt(x^3+5)))^2)\ dx`

`int_0^2sqrt(1+(9x^4)/(4(x^3+5)))\ dx`

This function doesn't have an elementary anti-derivative (that I've been able to find). The best we can really do is use an approximation. Here I did a midpoint Riemann rectangle sum with 25 rectangles to get an answer of about `2.58`: