Question

Consider the given functions f(x)=x and g(x)=x^{2}-6 To find the area between f(x) and g(x) complete the given integral:______ { Area }=μA}=3 \

Answer 1

To find the area between the functions f(x) = x and g(x) = x^2 - 6, set the two equations equal to each other, find the x-values of intersection, and evaluate the integral from x = -2 to x = 3.

Step-by-step explanation:

To find the area between the functions f(x) = x and g(x) = x^2 - 6, we need to find the x-values where the two functions intersect. By setting the two equations equal to each other, we get x = x^2 - 6. Rearranging, we have x^2 - x - 6 = 0, which can be factored as (x - 3)(x + 2) = 0. So the two x-values of intersection are x = 3 and x = -2.

The area between the two functions is given by the definite integral from x = -2 to x = 3 of (f(x) - g(x)) dx. Plugging in the equations, we have the integral of (x - (x^2 - 6)) dx. Evaluating this integral, we find the area is 3 square units.

Therefore, the completed integral is:

Area = 3