Question

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
ANSWERED
asked 2021-01-24
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 \(\int_{0}^{2}\ f(x)\ dx?\)

Answers (1)

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
\(= \int_{0}^{2}\ f\ (x)\ dx\)
\(= \int_{0}^{2} \left[\frac{x}{x}\ +\ \frac{1}{x}\right]dx\)
\(= \int_{0}^{2} (1\ +\ \frac{1}{x})\ dx\)
\(= [x\ +\ \ln\ x]_{0}^{2}\)
\(=(2\ +\ \ln\ 2)\ -\ (0\ +\ \ln\ 0)\)
\(= 2\ +\ \ln\ 2\)

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