Working with square roots

Question

I'm asking for methods on working with and dealing with square roots in algebra. Numerous times square roots have been the bane of my existence and are usually difficult to get around.

E.g. the other day, I was working on finding the distance between the point $x=-1$ and $x=1$ on the curve $x^2$, whilst tracing the path of the curve (not the straight distance). I eventually established that the distance can be given by:

$$\lim_{n\to \infty} \sum_{i=1}^n \frac{2}{n^2} \sqrt{16i^2+16in-16i+5n^2-8n+4}$$

Though I was unable to evaluate or even "play" with it due to the square root halting any of my attempts at evaluating the sum.

The square roots basically stop me in my tracks from even trying to play around with the question.

Are there any methods of getting rid of square roots, or at least making it significantly easier to deal with?

Answer 1

\begin{align*} \frac{2\sqrt{16k^2+16nk-16k+5n^2-8n+4}}{n^2} &= \frac{2}{n} \sqrt{5+\frac{16k-8}{n}+\frac{4(2k-1)^2}{n^2}} \\ &= \frac{2}{n} \sqrt{5+16\left( \frac{2k-1}{2n} \right)+ 16\left( \frac{2k-1}{2n} \right)^2} \\ \end{align*}

The Riemann sum is mid-point rule of:

$$\int_{0}^{1} 2\sqrt{5+16x+16x^2} dx= \frac{6\sqrt{37}-2\sqrt{5}+\sinh^{-1}6-\sinh^{-1}2}{4}$$