Question

Find the area of the region bounded by the graphs of the equations \(y=\cos ^{2} x, y=\sin ^{2} x, x=-\frac{\pi}{4}\), and \(x=\frac{\pi}{4}\).

Short answer

The area of the region is 1.

Step-by-step solution

Identify Intersection Points

First, we need to find the points where the graphs of the functions intersect when bounded by the lines. We set the functions equal: \[\cos^2 x = \sin^2 x.\]Rewriting using the identity \( \cos^2 x + \sin^2 x = 1 \), this simplifies to \( \cos^2 x = 1/2 \). Solving gives the known angles \( x = -\frac{\pi}{4} \) and \( x = \frac{\pi}{4} \).

Set Up the Definite Integral

Next, we determine which function is on top and which is on the bottom between the points of intersection by considering an interval, e.g., from \( x = -\frac{\pi}{4} \) to \( x = \frac{\pi}{4} \). We observe that \( \cos^2 x = \sin^2 x \) at these bounds and use the symmetry property of these functions over the interval. We set up the integral: \[\int_{-\pi/4}^{\pi/4} (\cos^2 x - \sin^2 x) \, dx.\]

Simplify Using Trigonometric Identities

To simplify the integration, we use the identity \( \cos^2 x - \sin^2 x = \cos 2x \). Therefore, the integral becomes: \[\int_{-\pi/4}^{\pi/4} \cos 2x \, dx.\]

Evaluate the Integral

The integral of \( \cos 2x \) is \( \frac{1}{2} \sin 2x \). Evaluate this from \( x = -\frac{\pi}{4} \) to \( x = \frac{\pi}{4} \):\[\left[ \frac{1}{2} \sin 2x \right]_{-\pi/4}^{\pi/4} = \frac{1}{2} \sin (\frac{\pi}{2}) - \frac{1}{2} \sin (-\frac{\pi}{2}).\]

Simplify and Calculate the Result

We know that \( \sin (\frac{\pi}{2}) = 1 \) and \( \sin (-\frac{\pi}{2}) = -1 \). So, the evaluated integral is:\[\frac{1}{2} (1) - \frac{1}{2} (-1) = \frac{1}{2} + \frac{1}{2} = 1.\]

Conclusion

Since all steps confirm the calculations, the area of the region is 1.

Key concepts

Trigonometric Identities

Trigonometric identities are special equations involving trigonometric functions that hold true for all values of the involved variables. They are extremely useful for simplifying complex expressions and solving equations. In this problem, we use the Pythagorean identity:

• \( \cos^2 x + \sin^2 x = 1 \)

This identity helps in expressing one trigonometric function in terms of another. For example, knowing that \( \cos^2 x = \sin^2 x \) can be rewritten using the identity to simplify computations significantly. This simplification is crucial when setting up problems like calculating areas or intersections of graphs.

Definite Integral

A definite integral calculates the accumulated "net area" between the graph of a function and the x-axis, from one point to another. This concept is important in determining the "size" of regions in many applied problems. To compute the area between two curves, you set up a definite integral, subtracting the lower curve from the upper curve within the limits:

• \( \int_{a}^{b} [f(x) - g(x)] \, dx \)

The symbols \( a \) and \( b \) are the lower and upper bounds, respectively. In our exercise, \( -\pi/4 \) and \( \pi/4 \) are the bounds, and we needed to figure out which function is on top or bottom within this range before integrating.

Cosine and Sine Functions

The cosine and sine functions are fundamental trigonometric functions that describe relationships in right-angled triangles. They also represent periodic oscillations. For example:

• The cosine function starts at a maximum value and decreases.
• The sine function starts at zero and increases to a maximum.

In the exercise, we focus on the squares of these functions, \( \cos^2 x \) and \( \sin^2 x \), which also appear frequently in various scientific calculations. By squaring these functions, their range becomes between 0 and 1, which affects their intersection points and area calculations. Understanding their cyclical nature and symmetry is crucial when determining how they interact over specific intervals.

Intersection Points of Functions

Intersection points are where two or more graphs meet. Finding these points is essential for solving many mathematical problems, such as calculating the area between curves. To find these points, set the functions equal and solve the resulting equation. This often requires algebraic manipulation or knowing specific identities. In this problem, we used:

• Set \( \cos^2 x = \sin^2 x \) and simplify using identities.
• Determine the known solutions for our specific interval \( x = -\pi/4 \) to \( x = \pi/4 \).

Intersection points also help in determining which function is the "upper" or "lower" boundary within the interval, which directly affects how we calculate areas with definite integrals. Recognizing these intersections aids in visualizing how the functions behave.