Final answer:
The approximate area enclosed by the curves can be calculated by integrating each function from x = 0 to x = 3 and subtracting the areas. Numerical approximation methods may be used due to the complexity of the functions.
Step-by-step explanation:
The approximate area enclosed by the curves y = \frac{x}{\sqrt{1+x^2}} and y = \frac{x}{\sqrt{9-x^2}} over the interval [0,3] can be found by integrating these functions with respect to x between the limits of 0 and 3. To calculate the total area between these two curves, we need to integrate each function separately and then subtract the area of the lower curve from the area of the upper curve. The integral of each function will give us the area under that curve from x = 0 to x = 3. After calculating each area, subtract the smaller from the larger to find the enclosed area.
Given the complexity of these functions, numerical methods or approximation techniques may be employed to estimate the integrals, as the antiderivatives of these functions are not elementary. Thus, a numerical approximation would involve dividing the interval into small segments, calculating the approximate area of each segment, and then summing these to find the total area.