find the area of the region under the curve y=f(x)=e^2x
over the indicated interval x<2
Int(e^2x dx) = 1/2 e^2x
1/2 (e^4 - e^-oo) = 1/2 e^4
To find the area of the region under the curve y = f(x) = e^(2x) over the interval x < 2, we need to evaluate the definite integral of f(x) over that interval.
The definite integral of a function f(x) over an interval [a, b] is denoted as ∫[a,b] f(x) dx. In this case, our interval is x < 2, so we need to find the integral from -∞ to 2.
To calculate this integral, follow these steps:
1. Determine the indefinite integral of f(x) = e^(2x). The indefinite integral of e^(2x) is given by: ∫ e^(2x) dx = (1/2)e^(2x) + C, where C is the constant of integration.
2. Evaluate the definite integral over the given interval. Since the upper bound is 2, the area under the curve is given by: ∫[-∞,2] e^(2x) dx = [(1/2)e^(2x)] between -∞ and 2.
To evaluate this integral, substitute the upper limit into the indefinite integral: [(1/2)e^(2*2)] - [(1/2)e^(2*(-∞))].
Since e^(-∞) is approaching 0 (since raised to any negative power), we simplify the expression to: [(1/2)e^4] - [(1/2)*0] = (1/2)e^4.
So, the area of the region under the curve y = e^(2x) over the interval x < 2 is (1/2)e^4.