Problem 38 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(37-42,\) sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\sin x, g(x)=\cos 2 x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The area can be calculated by first drawing the graphs of the functions, finding the intersection points, setting up the integral over these intersection points taking the absolute value of the difference between the functions, and then evaluating this integral.

Step 1: Draw the Graphs and Find Intersection Points

First, the graphs of the two functions \(f(x) = \sin x\) and \(g(x) = \cos 2x\) are sketchd in the given interval. To find the intersection points, set the two functions equal to each other, i.e., \(\sin x = \cos 2x\) and then solve for \(x\) over the interval \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{6}\).

Step 2: Setup the Integral

After getting the intersection points, which are the limits of the integral, setup the integral to calculate the area. As the area should always be positive, the absolute value is used in the differences between the two functions. So, the area \(A\) is given by \(A = \int_a^b |f(x) - g(x)| dx\).

Step 3: Calculate the Integral

Evaluate the integral over the interval from the smaller to the larger intersection points, this will give the area object to the problem.

Key Concepts

Understanding Integral Calculus

Integral calculus is a branch of mathematics that focuses on accumulations, such as areas under curves or the collection of quantities moving through an amount of space. It is the counterpart of differential calculus, which is about rates of change.
To find the area between two curves, we use the definite integral, which sums up infinite small slices of area between two points on a graph.
This process involves:

• Sketching the curves to perceive the problem visually.
• Finding intersection points to set limits for integration.
• Using integrals to calculate the area, ensuring it is positive.

Tackling these steps individually helps translate the complex graphical problems into manageable mathematical operations.

Finding Intersection Points

Intersection points are where two graphs intersect each other. Identifying these points is crucial when calculating the area between curves.
For trigonometric functions like \(f(x) = \sin x\) and \(g(x) = \cos 2x\), we set the equations equal to solve for \(x\).
This involves solving \(\sin x = \cos 2x\) within a given interval, for example, \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{6}\).

• Equate the functions: \(\sin x = \cos 2x\).
• Solve the equation using trigonometric identities or algebraic manipulation to find exact or approximate \(x\) values.
• These \(x\) values become the integration limits.

Intersection points ensure that you integrate within the correct boundaries, as they indicate where one function transitions from being above or below the other.

Exploring Trigonometric Functions

Trigonometric functions like \(\sin x\) and \(\cos 2x\) represent periodic oscillations that alternate between -1 and 1.
They are useful in modeling cyclic phenomena, such as waves or circular motion.
Key points in trigonometric functions include:

• Period: The distance over which a function completes one full cycle. \(\sin x\) has a period of \(2\pi\), whereas \(\cos 2x\) has a shorter period of \(\pi\).
• Amplitude: The height from the center line to the peak, typically 1 for basic functions like \(\sin\) and \(\cos\).
• Functions can shift vertically, horizontally, or stretch and shrink based on the equation.

Understanding these aspects allows you to better sketch and interpret the graphs, particularly when defining the area concerning other functions.

Definite Integral and Area Calculation

The definite integral is a tool used to find the total accumulation between a function and the x-axis over a specific interval.
When finding the area between two curves, this principle is adapted slightly.

• Calculate the integral \(\int_a^b (f(x) - g(x)) \, dx\) to find the total area.
• Since area should be positive, ensure to use absolute values if necessary: \(\int_a^b |f(x) - g(x)| \, dx\).
• This accounts for cases where the graphs swap positions over the interval.

By evaluating the definite integral from one intersection point to the other, you effectively sum the series of vertical slices that fill the region between the two curves.