Arc Length of $f(x)$ on [0,a]

[karush]

Find the arc length of
$$f (x)=(1/3)(x^2 +2)^{3/2}$$
On the interval [0, a]

The parametric I got

$$y=t$$
$$x=\sqrt{(3t)^{2/3}-2}$$

I proceeded but didnt get the answer of

$$a+\frac{a^3}{3}$$

 

[MarkFL]

Re: Lenght of a curve

I wouldn't bother with parametrization...:)

We have:

\(\displaystyle f(x)=\frac{1}{3}\left(x^2+2\right)^{\frac{3}{2}}\)

Hence:

\(\displaystyle f'(x)=x\sqrt{x^2+2}\)

And so the arc-length $s$ will be given by:

\(\displaystyle s=\int_0^a\sqrt{1+\left(x\sqrt{x^2+2}\right)^2}\,dx\)

Can you proceed?

 

[karush]

$$\displaystyle
s=\int_0^a\sqrt{1+\left(x\sqrt{x^2+2}\right)^2}\,dx
=\int_{0}^{a} \left({x}^{2}+1\right)\,dx
=\frac{a^3}{3}+a
$$

When do we use parametrics for length of curve