Some of the limits are easy to see, but I want to use Riemman integral to determine the value of the following limits.
$$\lim\_{n\to\infty} (\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n})$$
$$\lim\_{n\to\infty} \frac{1}{n} \sqrt[n]{(n+1)(n+2)\cdots(n+n)}$$
$$\lim\_{n\to\infty} (\sin {\frac{n}{n^2+1^2}}+\sin {\frac{n}{n^2+2^2}}+\cdots+\sin {\frac{n}{n^2+n^2}})$$

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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.