14–25. {Use of Tech} Areas of regions Determine the area of the given region.

Verified step by step guidance

1. Identify the curves given: the upper curve is \(y = \sin 2x - 1\) and the lower curve is \(y = -\cos 2x\) over the interval where they intersect.
2. Find the points of intersection by setting \(\sin 2x - 1 = -\cos 2x\) and solving for \(x\). This will give the limits of integration.
3. Set up the integral for the area of the shaded region as the integral of the difference between the upper and lower curves: \(\int_{a}^{b} \left[(\sin 2x - 1) - (-\cos 2x)\right] \, dx\) where \(a\) and \(b\) are the intersection points.
4. Simplify the integrand to \(\int_{a}^{b} (\sin 2x - 1 + \cos 2x) \, dx\) to prepare for integration.
5. Integrate the function term-by-term with respect to \(x\), then evaluate the definite integral using the limits \(a\) and \(b\) to find the area.

Key Concepts

Finding Points of Intersection

To determine the area between two curves, first find their points of intersection by setting the functions equal to each other. These points serve as the limits of integration, defining the interval over which the area is calculated.

Definite Integral for Area Between Curves

The area between two curves y = f(x) and y = g(x) over an interval [a, b] is found by integrating the difference f(x) - g(x) with respect to x. This integral sums the vertical distances between the curves across the interval.

Trigonometric Functions and Their Properties

Understanding the behavior of trigonometric functions like sin(2x) and cos(2x) is essential, including their periodicity and transformations. This knowledge helps in accurately setting up the integral and interpreting the graph.