Final answer:
To find the area of the region bounded by the given equations y = -x - x^2 and y = -6, we can determine the points of intersection and calculate the area between those points. The points of intersection are x = -3 and x = 2. By evaluating the definite integral of the difference between the two equations from x = -3 to x = 2, we find that the area is 46/3 or 15.33 units squared.
Step-by-step explanation:
To find the area of the region bounded by the given equations, we need to determine the points of intersection between the curves and calculate the area between those points.
First, let's find the points of intersection between the equations y = -x - x^2 and y = -6. Setting the two equations equal to each other, we have:
-x - x^2 = -6
Simplifying, we get x^2 + x - 6 = 0. Factoring, we have (x + 3)(x - 2) = 0. This gives us two solutions: x = -3 and x = 2.
Next, we need to find the y-values for each x-value. Substituting x = -3 into either equation gives y = -3, and substituting x = 2 gives y = -10.
The area between the curves can be found by calculating the definite integral of the difference between the two equations, from x = -3 to x = 2:
A = ∫ (y₂ - y₁) dx = ∫ (-6 - (-x - x^2)) dx = ∫ (-6 + x + x^2) dx = [-6x + (x^2/2) + (x^3/3)]|-32
Substituting the values of x, we get:
A = [-6(2) + (2^2/2) + (2^3/3)] - [-6(-3) + ((-3)^2/2) + ((-3)^3/3)]
Finally, simplifying this gives the area as A = 46/3 or 15.33 units squared.