Find the area of the shaded region enclosed by the following functions.
\[
\begin{array}{l}
y=\frac{9}{2} x \\
y=9 \\
y=\frac{9}{49} x^{2}
\end{array}
\]
Set up the integral that gives the area of the shaded region.

Step 1 ：First, we need to find the points of intersection between the functions. This can be done by setting the functions equal to each other and solving for x. The points of intersection will be the limits of integration.

Step 2 ：The upper function will be the constant function \(y=9\) and the lower function will be the maximum of the other two functions. We can find which function is the lower one by comparing the functions at a point between the points of intersection.

Step 3 ：After setting up the integral, we can calculate the area by evaluating the integral.

Step 4 ：By solving, we find that the points of intersection are \(x = -7, 0, 2\).

Step 5 ：Comparing the functions at a point between the points of intersection, we find that the lower function is \(y = 4.5x\) for \(x < 0\) and \(y = 0.183673469387755x^2\) for \(x > 0\).

Step 6 ：Setting up the integral, we find that the area of the shaded region is given by the integral of \(9 - 4.5x\) from \(x = -7\) to \(x = 0\) plus the integral of \(9 - 0.183673469387755x^2\) from \(x = 0\) to \(x = 2\).

Step 7 ：Evaluating the integral, we find that the area of the shaded region is \(56.25\).

Step 8 ：Final Answer: The area of the shaded region enclosed by the functions \(y=\frac{9}{2} x\), \(y=9\), and \(y=\frac{9}{49} x^{2}\) is \(\boxed{56.25}\).