Arc Length Exercises I am studying for a repeat exam.
What are examples of functions that give an easy integral once I apply the arc length formula?
Thank you very much if you can help.
 A: Here are some tips.
First you can google "arc length problems calculus 2" and you will get lots of examples through various websites. Don't look ow they work it out, you first try yourself.
Second, Stewart Calculus has a lot of arc length problems (current edition chapter 8 section 1). You can buy this book used for a good price. Excellent reference material. 
I am sure there are other Calculus books in circulation that also have enough arc length problems. When I was studying it, I used Thomas & Finney as well as Edwards & Penney for reference. (Overseas editions)
A: Almost any book has enough practic3e for your purposes. There are very few really different functions $y=f(x)$ for which the integral you end up with is "doable." This is because if you take most $f(x)$, and compute
$$\sqrt{1+(f'(x))^2},\tag{1}$$ 
you end up with something that cannot be integrated in elementary terms. That's why in many of the problems you are asked to do, the function $1+(f'(x))^2$ "magically" ends up being the square of something nice. 
A typical example would be something like $f(x)=\frac{1}{2}(e^x+e^{-x})$. Differentiate. We get $\frac{1}{2}(e^x-e^{-x})$. Square and add $1$. We get 
$\frac{1}{4}(e^{2x}-2+e^{-2x})+1$. 
When you bring the expression to a common denominator, you end up with $\frac{1}{4}(e^{2x}+2+e^{-2x})$, which "just happens to be" the square of $\frac{1}{2}(e^x+e^{-x})$. So take the square root, and now the integration is easy.
In arclength problems, it is useful to be on the lookout for magic simplifications.  The unfortunate thing is that if you make a minor mistake in the algebra, you end up with something that cannot be integrated in elementary terms.
