Question

(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

Answer 1

A = 0 b = pi/8 delta x = b - a/n = pi/8 - 0/n = pi/8n x_1 = a + I delta x = 0 + I pi/8n = (I pi/8n) hence f (x) = tan x on the interval [0, pi/8]