To find the area under the curve \(y = x^3 - 2x^2 + 5x\) from \(x = 0\) to \(x = 6\), we need to integrate the function with respect to \(x\).

The integral of \(y = x^3 - 2x^2 + 5x\) is:

\(\int (x^3 - 2x^2 + 5x) \, dx = \frac{x^4}{4} - \frac{2x^3}{3} + \frac{5x^2}{2} + C\)

We evaluate this definite integral from \(x = 0\) to \(x = 6\):

\(\left[ \frac{x^4}{4} - \frac{2x^3}{3} + \frac{5x^2}{2} \right]_0^6\)

Substitute \(x = 6\):

\(\frac{6^4}{4} - \frac{2 \times 6^3}{3} + \frac{5 \times 6^2}{2} = \frac{1296}{4} - \frac{2 \times 216}{3} + \frac{5 \times 36}{2}\)

\(= 324 - 144 + 90\)

\(= 270\)

Thus, the area of the region is 270.