Determine the area of the shaded region in the following figures.

Verified step by step guidance

1. Step 1: Identify the curves that bound the shaded region. From the graph, the shaded region is enclosed between the curves x = y^2 - 3y + 12 (black curve) and x = -2y^2 - 6y + 30 (red curve).
2. Step 2: Determine the limits of integration. The shaded region spans from y = -2 to y = 2, as indicated by the intersection points of the two curves along the y-axis.
3. Step 3: Set up the integral for the area. The area of the shaded region is given by the integral of the difference between the rightmost curve (x = -2y^2 - 6y + 30) and the leftmost curve (x = y^2 - 3y + 12) with respect to y. The integral is: ∫[-2 to 2] [(-2y^2 - 6y + 30) - (y^2 - 3y + 12)] dy.
4. Step 4: Simplify the integrand. Combine like terms to simplify the expression inside the integral: (-2y^2 - 6y + 30) - (y^2 - 3y + 12) = -3y^2 - 3y + 18.
5. Step 5: Evaluate the integral. Integrate the simplified expression -3y^2 - 3y + 18 with respect to y over the interval [-2, 2]. This involves finding the antiderivative and substituting the limits of integration.

Key Concepts

Definite Integral

A definite integral calculates the net area under a curve between two specified points on the x-axis. It is represented as ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. This concept is crucial for finding the area of the shaded region between two curves, as it allows us to quantify the space enclosed by them.

Intersection Points

Intersection points are the values of x where two curves meet, which are essential for determining the limits of integration in area calculations. To find these points, one must set the equations of the curves equal to each other and solve for x. Identifying these points helps in accurately defining the boundaries of the shaded region whose area we want to calculate.

Area Between Curves

The area between two curves is found by integrating the difference of the functions that define the curves over the interval defined by their intersection points. If f(x) is the upper curve and g(x) is the lower curve, the area A can be expressed as A = ∫[a, b] (f(x) - g(x)) dx. This concept is fundamental for solving the problem of finding the area of the shaded region in the given figure.