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$$x= y^{1/30},\; 0 ≤ y ≤ 2$$

I know the formula we use is $\sqrt{1 +f'(x)^2} dx. $

But now do I switch the function so it's "y="?

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Take each side to a power of $30$: $$x^{30} = \left(y^{1/30}\right)^{30} = y$$

$$y = x^{30} \implies y' = 30x^{29} \implies (y')^2 = \Big(30x^{29}\Big)^2$$

That gives us the length of the curve, $s$:

$$s =\int_0^2\sqrt{1 + \Big(30x^{29}\Big)^2}\;dx$$

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You don't need to. Just treat X as a function of y. And use dy instead. It is just nomenclature; how stuff is named.

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