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

Verified step by step guidance

1. Identify the curves that bound the shaded region. The upper curve is the parabola y = x^2 - 4x, and the lower curve is the line y = -2x.
2. Find the points of intersection of the two curves by setting x^2 - 4x = -2x. Solve this equation to determine the x-coordinates of the intersection points.
3. Set up the integral to calculate the area between the curves. The area is given by the integral of the difference between the upper curve and the lower curve, ∫[(x^2 - 4x) - (-2x)] dx, over the interval determined by the intersection points.
4. Simplify the integrand to combine like terms. The integrand becomes x^2 - 2x, so the integral to evaluate is ∫(x^2 - 2x) dx over the interval.
5. Evaluate the definite integral by finding the antiderivative of x^2 - 2x, applying the Fundamental Theorem of Calculus, and substituting the limits of integration (the x-coordinates of the intersection points).

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 upper function (the parabola) and the lower function (the line) over the interval where they intersect.

Finding Intersection Points

To determine the area between two curves, it is essential to find their intersection points, as these points define the limits of integration. This involves solving the equations of the curves simultaneously to find the x-values where they intersect. In this case, solving the equations of the parabola and the line will yield the bounds for the definite integral.

Area Between Curves

The area between two curves is calculated by integrating the difference of the functions that define the curves over the interval determined by their intersection points. The formula for this area is given by ∫[a, b] (f(x) - g(x)) dx, where f(x) is the upper function and g(x) is the lower function. This concept is crucial for accurately determining the area of the shaded region in the provided graph.