# Suppose that the n-th Riemann sum (with n sub-intervals of equal length), for some function f(x) on the intrval [0, 2], is S_{n} = frac{2}{n^{2}} times sum_{k = 1}^{n} k. What is the value of int_{0}^{2} f(x) dx?

Confidence intervals
Suppose that the n-th Riemann sum (with n sub-intervals of equal length), for some function f(x) on the intrval [0, 2], is $$S_{n} = \frac{2}{n^{2}}\ \times\ \sum_{k = 1}^{n}\ k.$$
What is the value of $$\int_{0}^{2}\ f(x)\ dx?$$

2021-01-25

Step 1
$$S_{n} = \frac{2}{n^{2}}\ \sum_{k = 1}^{n}\ k$$
$$S_{n} = \frac{2}{n^{2}\ [1\ +\ 2\ +\ 3\ +\ \cdots\ +\ n]}$$
$$S_{n} = \frac{2}{n^{2}}\frac{n(n\ +\ 1)}{2}$$
$$S_{n} = \frac{n\ +\ 1}{n}$$
$$f(x) = \frac{x\ +\ 1}{x}$$
Step 2
Now to find
$$= \int_{0}^{2}\ f\ (x)\ dx$$
$$= \int_{0}^{2} \left[\frac{x}{x}\ +\ \frac{1}{x}\right]dx$$
$$= \int_{0}^{2} (1\ +\ \frac{1}{x})\ dx$$
$$= [x\ +\ \ln\ x]_{0}^{2}$$
$$=(2\ +\ \ln\ 2)\ -\ (0\ +\ \ln\ 0)$$
$$= 2\ +\ \ln\ 2$$