Question

r is the region bounded by the functions f(x)=2ex−1 and g(x)=x2−2. find the area of the region bounded by the functions on the interval [−1,1]. enter an exact answer.

Answer 1

To find the area of the region bounded by the functions f(x) = 2ex-1 and g(x) = x^2-2 on the interval [-1,1], we need to find the points of intersection and perform integration. The area is 2e.

Step-by-step explanation:

To find the area of the region bounded by the functions f(x) = 2ex-1 and g(x) = x^2-2 on the interval [-1,1], we need to find the points of intersection between the two functions. Setting f(x) = g(x), we get 2ex-1 = x^2-2. Solving this equation gives x = -1 and x = 1 as the points of intersection.

Next, we integrate f(x) - g(x) from x = -1 to x = 1 to find the area between the two curves. Integrating 2ex-1 - (x^2-2) with respect to x, we get the area as A = ∫(2ex-1 - (x^2-2)) dx from -1 to 1.

Performing the integration gives A = (2e - 2) - (-2) = 2e - 2 + 2 = 2e.