To find the area of the shaded region, we need to integrate the function \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) from \(x = 0\) to \(x = 4\).

The integral is:

\(\int_{0}^{4} \left( 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}} \right) \, dx\)

Integrating term by term, we have:

\(\int 10x^{\frac{1}{2}} \, dx = \frac{10}{\frac{3}{2}} x^{\frac{3}{2}} = \frac{20}{3} x^{\frac{3}{2}}\)

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

Thus, the integral becomes:

\(\left[ \frac{20}{3} x^{\frac{3}{2}} - x^{\frac{5}{2}} \right]_{0}^{4}\)

Evaluating at the bounds:

\(\left( \frac{20}{3} \times 8 - 32 \right) - \left( 0 \right) = \frac{160}{3} - 32\)

\(= \frac{160}{3} - \frac{96}{3} = \frac{64}{3}\)

Therefore, the area of the shaded region is \(\frac{64}{3}\) or approximately \(21.3\).