approximate value of the area of the region y=4x^(2),y=e^(x)

Step-by-step solution:

1. Set 4x^2 = e^x to find intersection points.

2. Solve for x using LambertW function: x = -2\text{LambertW}(-\frac{1}{4}) and x = -2\text{LambertW}(\frac{1}{4}) .

3. Determine that 4x^2 is on top in the interval between intersection points.

4. Set up the integral A = \int_{-2\text{LambertW}(-\frac{1}{4})}^{-2\text{LambertW}(\frac{1}{4})} (4x^2 - e^x) \, dx .

5. Integrate and evaluate at the limits.

6. Numerically calculate the area to get approximately 0.801 square units.