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

Verified step by step guidance

1. Step 1: Identify the equations of the curves that bound the shaded region. In this case, the parabola is given by y = x^2 - 2, and the line is given by y = x.
2. Step 2: Determine the points of intersection between the parabola and the line by solving the equation x^2 - 2 = x. Rearrange to form a quadratic equation: x^2 - x - 2 = 0.
3. Step 3: Solve the quadratic equation x^2 - x - 2 = 0 using factoring, the quadratic formula, or another method. The solutions will give the x-coordinates of the points of intersection.
4. Step 4: Set up the integral to calculate the area of the shaded region. The area is found by integrating the difference between the upper curve (y = x) and the lower curve (y = x^2 - 2) over the interval determined by the points of intersection.
5. Step 5: Evaluate the integral ∫[x1 to x2] (x - (x^2 - 2)) dx, where x1 and x2 are the x-coordinates of the points of intersection. Simplify the integrand and compute the integral step by step to find the area.

Key Concepts

Definite Integral

The definite integral is a fundamental concept in calculus that represents the signed area under a curve between two points on the x-axis. It is calculated using the integral symbol and provides a way to quantify the area of regions bounded by curves, such as the parabola and the line in the given problem. The definite integral is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the limits of integration.

Finding Intersection Points

To determine the area between two curves, it is essential to find their intersection points, where the curves meet. These points are found by setting the equations of the curves equal to each other and solving for x. In this case, the intersection points of the line y = x and the parabola y = x² - 2 will define the limits of integration for calculating the area of the shaded region.

Area Between Curves

The area between two curves can be calculated by integrating the difference of the functions that define the curves over the interval defined by their intersection points. Specifically, 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 method allows for the accurate calculation of the area of the shaded region in the provided graph.