Final answer:
To determine the area between the curves y=-x and x=y²-6, find the intersection points by solving the equations, and then integrate the difference between the functions over the interval given by the intersection points to find the enclosed area.
Step-by-step explanation:
The question asks to find the area between the curves y=-x and x= y²-6. To determine this area, we need to find the points of intersection of the two curves and integrate the difference of the functions within the range of those intersection points. The intersection points are found by setting the two equations equal to each other and solving for the variables.
First, we substitute y from the equation y=-x into the second equation, yielding x=(-x)²-6, which simplifies to x=x²-6. Solving this quadratic equation will give us the x-values of the points of intersection. Next, we can integrate the difference between the two functions over the interval defined by these x-values to find the enclosed area. The integral calculation involves finding the integral of the upper function minus the lower function from the left intersection point to the right one.
If both intersection points are above the x-axis, this will give us part (a) - the area between the curves above the x-axis. If both are below, then part (b) - the area below. If one point is above and one is below, or if the curves intersect at more than two points, we will have to divide the problem accordingly and calculate the areas separately to find part (c) - the area enclosed by the curves. In the case there are no real intersections, part (d) would be correct, implying no area between the curves exists.