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 curves are y = x (a straight line) and y = x^2 - 2 (a parabola).
2. Step 2: Determine the points of intersection between the two curves by solving the equation x = x^2 - 2. Rearrange this into a standard quadratic form: x^2 - x - 2 = 0, and solve for x using factoring or the quadratic formula.
3. Step 3: Set up the integral to calculate the area of the shaded region. The area is given by the integral of 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.
4. Step 4: Write the integral expression for the area: A = ∫[x1 to x2] (x - (x^2 - 2)) dx, where x1 and x2 are the x-values of the points of intersection found in Step 2.
5. Step 5: Simplify the integrand to combine terms: A = ∫[x1 to x2] (-x^2 + x + 2) dx. Then, compute the integral term by term, applying the power rule for integration to each term. Finally, evaluate the definite integral by substituting the limits of integration (x1 and x2).

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area of the shaded region can be found by integrating the difference between the two functions that bound the region.

Intersection Points

Intersection points are the x-values where two functions meet, which are crucial for determining the limits of integration. In this case, finding where the line y = x intersects the parabola y = x² - 2 will provide the bounds for the area calculation. Solving the equation x = x² - 2 will yield these points.

Area Between Curves

The area between curves is calculated by integrating the top function minus the bottom function over the interval defined by their intersection points. In this scenario, the area of the shaded region is found by integrating the difference between the linear function y = x and the quadratic function y = x² - 2, from the left intersection point to the right intersection point.