# (2 points) The area A of the region S that lies under the graph of the...

(2 points) The area $$A$$ of the region $$S$$ that lies under the graph of the continuous function $$f$$ on the interval $$[a, b]$$ is the limit of the sum of the areas of approximating rectangles:

$$A=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x$$

where $$\Delta x=\frac{b-a}{n}$$ and $$x_{i}=a+i \Delta x$$.

The expression

$$A=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\pi}{8 n} \tan \left(\frac{i \pi}{8 n}\right)$$

gives the area of the function $$f(x)=$$ on the interval

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