Final answer:
To find the area between f(x)=2eˣ−2 and g(x)=x²−5 on the interval [−1,1], calculate the integral of the upper function minus the lower function, ∫_{-1}^{1} [2eˣ−2 - (x²−5)] dx, after determining the points of intersection and the respective positions of the functions over the interval.
Step-by-step explanation:
To find the area of the region bounded by the functions f(x)=2eˣ−2 and g(x)=x²−5 on the interval [−1,1], we need to set up an integral that represents the area between these two curves. We start by finding the points of intersection between the two functions over the interval by setting them equal to each other and solving for x. Then, we integrate the upper function minus the lower function across the interval of intersection.
Let's assume that the functions have been graphed, and we've found that f(x) lies above g(x) for the interval [−1,1]. The area A can be calculated as:
A = ∫_{-1}^{1} [f(x) - g(x)] dx = ∫_{-1}^{1} [2eˣ−2 - (x²−5)] dx
This integral can be evaluated using the antiderivative formulas for eˣ, x², and constants. After finding the antiderivatives, we plug in the limits of integration to find the numerical value of the area. If there are multiple regions where one function is above the other, we would calculate the area of each region separately and sum the absolute values to get the total area.