TITLE: Calculate limit using Riemann integral
QUESTION [5 upvotes]: Some of the limits are easy to see, but I want to use Riemman integral to determine the value of the following limits.
$$\lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n})$$
$$\lim_{n\to\infty} \frac{1}{n} \sqrt[n]{(n+1)(n+2)\cdots(n+n)}$$
$$\lim_{n\to\infty} (\sin {\frac{n}{n^2+1^2}}+\sin {\frac{n}{n^2+2^2}}+\cdots+\sin {\frac{n}{n^2+n^2}})$$

For exmaple in first, I suppose $\int_a^b f(x)dx$ is equal to the limit
$$\lim_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n}) = \lim_{n\to\infty} \sum_{i = 1}^{2n} \frac{1}{n+i} = \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)i}{2n})\cdot\frac{(b-a)i}{n}$$
But cannot find anything.  
Please don't tell me the method directly, I would appreciate it if you could give me some crucial hints for one\all of them. Thank you very much.

REPLY [2 votes]: The $3$rd limit is straightforward by Taylor series and then Riemann sum 
$$\lim_{n\to\infty}\sum_{k=1}^{n}\sin \frac{n}{n^2+k^2} = \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+(k/n)^2} + O\left(\lim_{n\to\infty}\sum_{k=1}^{n}\left(\frac{n}{n^2+k^2}\right)^3\right)=\int_0^1\frac{1}{1+x^2}=\frac{\pi}{4}$$