A curve \( C \) defined by the equation
\[
y=\frac{1}{8} x^{3}-2 \sqrt{x}-2 x+10, \quad x>0,
\]
encloses a region \( R \) with the vertical line \( x=1 \), and a tangent line at point \( P(4,6) \) on \( C \) as shown in the figure below:
The tangent line crosses the \( x \)-axis at point \( Q \). Find the value of the generated area \( R \), writing your answer to 2 decimal places.
A. 2.89
B. 3.35
C. 7.16
D. 5.23