Integral usando suma de riemann

Sea

f(x) = 2x^2 - 3x

Intervalo: vos pusiste [4,0], pero supongo que será [0,4]

Ahora vamos con los pasos de Riemman

$$\begin{align}&\Delta x=\frac{4}{n}\\&x_i = i \Delta x = \frac{4i}{n}\\&\sum_{i=1}^n f(x_i) \Delta x=\sum_{i=1}^n \bigg(2 (\frac{4i}{n}) ^2 - 3(\frac{4i}{n}) \bigg)\frac{4}{n}=\\&\sum_{i=1}^n \bigg(\frac{32i^2}{n^2} - \frac{12i}{n} \bigg)\frac{4}{n}=\\&\sum_{i=1}^n \frac{128i^2}{n^3} - \frac{48i}{n^2} =\\&\sum_{i=1}^n \frac{128i^2}{n^3} - \sum_{i=1}^n \frac{48i}{n^2} =\\&\frac{128}{n^3}\sum_{i=1}^n i^2 - \frac{48}{n^2} \sum_{i=1}^n i =\\&\frac{128}{n^3} (\frac{n(n+1)(2n+1)}{6}) - \frac{48}{n^2} (\frac{n(n+1)}{2}) =\\&\frac{64}{n^3} (\frac{2n^3+3n^2+n}{3}) - \frac{24}{n^2} (n^2+n) =\\&\text{Cuando n tiende a infinito...}\\&\frac{128}{3}-24=\frac{56}{3}\end{align}$$

Te dejo calcular la integral para verificar que coincide...

Salu2

Oh vale, 3x