Question

Express the given integral as the limit of a Riemann sum but do not evaluate: the integral from 0 to 3 of the quantity x cubed minus 6 times x, dx.

Answer 1

https://i.gyazo.com/84c7a6d224d5e4d998b58baf1ab6f90e.png

Answer 2

$$ \sum_{i=1}^{n} f(x_i^*) \Delta x_k$$

Answer 3

let \[ \Delta x = \frac{ b - a } {n} \]a=0 , b = 3, \[ \Delta x = \frac{ 3 - 0 } {n} = \frac{3}{n}\]\[ \lim_{ n \to \infty}~ \sum_{i=1}^{n} f(a + i \Delta x) \Delta x
\\ \lim_{ n \to \infty}~ \sum_{i=1}^{n} f\left( 0 + i \cdot \frac 3 n \right) \frac 3 n
\\ \lim_{ n \to \infty}~ \sum_{i=1}^{n} f\left( \frac {3i}{n} \right) \frac 3 n
\\ \lim_{ n \to \infty}~ \sum_{i=1}^{n} f\left( \frac {3i}{n} \right) \frac 3 n
\\ \lim_{ n \to \infty}~ \sum_{i=1}^{n} \left( \left( \frac {3i}{n} \right)^3 - 6\left( \frac {3i}{n} \right) \right) \frac 3 n
\\ \lim_{ n \to \infty}~ \sum_{i=1}^{n} \left( \frac {27i^3}{n^3} - \frac {18i}{n} \right) \frac 3 n


\]

Answer 4

I think thats fine, unless you want to simplify it further

Answer 5

Thanks