Question

Evaluate the Riemann sum for (x) = x3 − 6x, for 0 ≤ x ≤ 3 with six subintervals, taking the sample points, xi, to be the right endpoint of each interval. Give three decimal places in your answer.

Answer 1

Hello,

Reminders:
$\sum_{i=1}^{n} i= \dfrac{n(n+1)}{2}$
$\sum_{i=1}^{n} i^2= \dfrac{n(n+1)}{2}$
$\sum_{i=1}^{n} i^3= \dfrac{n^2(n+1)^2}{4}$

$n=6\\ x_{0}=0\\ x_{n}=3\\ \Delta= \dfrac{x_n-x_0}{n}=0.5$
$x_{i} =0+\Delta*i= \frac{i}{2}$

$\sum_{i=1}^{n}\ \Delta*f(x_{i})=\sum_{i=1}^{6}\ \dfrac{i^3/8-6i}{2}$
$= \dfrac{1}{2}\sum_{i=1}^{6} (\frac{i^3}{8} -6i)=\dfrac{1}{16}*\frac{(6*7)^2}{4}- \frac{9*7}{2}= -3.9575$