Problem 68 - Essential Calculus

Question

Find the area of the region bounded by the graphs of the equations. $$ y=\cos ^{2} x, \quad y=\sin x \cos x, \quad x=-\pi / 2, \quad x=\pi / 4 $$

Step-by-Step Solution

Answer

The area of the region bounded by the graphs of the equations can be found by evaluating the definite integral \[\int_{-\pi/2}^{\pi/4}\left(\frac{1+\cos 2x}{2}-\frac{1}{2}\sin 2x\right) dx\]. The resulting value represents the area.

Step 1: Rewrite the function

Rewrite the functions using trigonometric identities: \(y=\cos ^{2} x=\frac{1+\cos 2x}{2}\) and \(y=\sin x \cos x = \frac{1}{2}\sin 2x\). The new functions are easier to integrate and simplify.

Step 2: Find intersects

Set the two functions equal to each other to find x-values where the functions intersect within the range of \(x=[-π/2,π/4]\): \(\frac{1+\cos 2x}{2}=\frac{1}{2}\sin 2x\). Solve for x to find the intersection points.

Step 3: Integrate functions

Integrate the functions from the left intersection to the right intersection found in Step 2. Subtract the integral of the lower function (\(\frac{1}{2}\sin 2x\)) from the integral of the upper function (\(\frac{1+\cos 2x}{2}\)) to find the area between the curves: \[\int_{-\pi/2}^{\pi/4}\left(\frac{1+\cos 2x}{2}-\frac{1}{2}\sin 2x\right) dx\]

Step 4: Evaluate the integral

Evaluate the definite integral found in step 3, applying the fundamental theorem of calculus.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate various trigonometric functions. These identities can simplify complex trigonometric expressions, making calculations easier. For example, one commonly used identity is the double angle formula for cosine: \[\cos^2 x = \frac{1 + \cos 2x}{2}\]. This transforms the square of the cosine function into an expression that is more straightforward for integration. Another identity is the product-to-sum formula for sine and cosine: \[\sin x \cos x = \frac{1}{2} \sin 2x\]. These identities not only simplify expressions but allow easier operations like integration. Recognizing and applying these identities effectively can save time and clarify the process of solving integrals or finding areas between curves.

Definite Integrals

Definite integrals represent the total area under a curve within a specified interval on the x-axis. In simpler terms, it calculates the accumulation of values, providing a numerical result. When setting up a definite integral, it's essential to specify the interval over which to integrate. In this problem, the interval is from \(-\pi/2\) to \(\pi/4\). The lower bound is \(-\pi/2\), and the upper bound is \(\pi/4\). Solving a definite integral involves:

• Finding the antiderivative of the function
• Evaluating it at both the upper and lower bounds
• Subtracting the value at the lower bound from the value at the upper bound

This process follows the fundamental theorem of calculus, connecting derivatives with integrals and making it possible to calculate exact areas.

Area Between Curves

Finding the area between two curves involves calculating the region enclosed by their graphs. The typical approach requires integrating the difference between two functions over a specific interval. The upper function usually indicates the top boundary of the area, while the lower function defines the bottom boundary.For this exercise, we encounter two functions:

• Upper function: \(y = \frac{1 + \cos 2x}{2}\)
• Lower function: \(y = \frac{1}{2} \sin 2x\)

To find the area between them, integrate the difference from the left intersection point to the right intersection point of the functions within the given interval: \[\int_{-\pi/2}^{\pi/4}\left(\frac{1 + \cos 2x}{2} - \frac{1}{2} \sin 2x\right) dx\]This integral essentially calculates the total vertical distance between the two curves across the specified range. Evaluating this definite integral provides the precise area encased by the two graphs within that interval.