极限 + 求和 => 积分
例：求 $\lim _{n \to\infty} \frac {\sum_{k=1}^{n} \sqrt {n^{2}-k^{2}}}{n^{2}}$
利用性质
$$
\int_0^a f(x) dx \approx \sum_{k=0}^{na} \frac{1}{n}f\left(\frac{k}{n}\right)\\
$$
原极限化为
$$
\frac{1}{n} \cdot\left(\sum_{k=1}^{n} \sqrt{1-\left(\frac{k}{n}\right)^{2}}\right)=\int_{0}^{1} \sqrt{1-x^{2}}
$$
$y^{2}=1-x^{2}$ $\Rightarrow x^{2}+y^{2}=1$
例：求 $\lim\limits_{n \to \infty}\sum\limits_{k=0}^n \dfrac {\sqrt {n}}{n+k^2}(n=1,2,\cdots)$
$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n} \frac{\sqrt{n}}{1+\frac{k}{\sin } j^{2}}$
let $t=\sqrt{n}$
$=\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n} \frac{t}{1+\left(\frac{k}{4}\right)^{2}}$ $=\lim _{n \rightarrow \infty} \frac{1}{t} \sum_{k=0}^{n} \frac{1}{1+\left(\frac{k}{4}\right)^{2}}$ $=\int_{0}^{1} \frac{dx^2}{1+x^{2}}=2 \arctan (1)=\frac{\pi}{4} \times 2=\frac{\pi}{2}$