Find the area under the given curve over the indicated interval. \[ y=e^{2 x} ;[0,2] \] The area is (Round to three decimal places as needed.)

Step 1 ：The problem is to find the area under the curve defined by the function \(y=e^{2x}\) over the interval [0,2].

Step 2 ：The area under a curve from a to b is given by the definite integral from a to b of the function.

Step 3 ：In this case, we need to find the definite integral from 0 to 2 of the function \(y=e^{2x}\).

Step 4 ：The result of the integration is \(-\frac{1}{2} + \frac{e^{4}}{2}\).

Step 5 ：Final Answer: The area under the curve is \(\boxed{-\frac{1}{2} + \frac{e^{4}}{2}}\).