Problem 37 - 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. $$ f(x)=\cos x, \mathrm{g}(x)=2-\cos x, 0 \leq x \leq 2 \pi $$

Step-by-Step Solution

Answer

The area between the curves \(f(x) = \cos x\) and \(g(x) = 2 - \cos x\) from \(0\) to \(2\pi\) is \(4\pi.\)

Step 1: Identify the Region

First, plot the two functions \(f(x) = \cos x\) and \(g(x) = 2 - \cos x\) on the same set of axes from \(0\) to \(2 \pi\). Identify the enclosed region between the two curves. Notice that \(g(x)\) is always above \(f(x)\) in this interval.

Step 2: Set Up the Integral

To find the area between the two curves from \(0\) to \(2\pi\), set up the definite integral \(A = \int_{0}^{2\pi} [g(x) - f(x)] dx\), where \(A\) is the area.

Step 3: Substitute the Functions

Substitute the actual functions to get: \(A = \int_{0}^{2\pi} [(2 - \cos x) - (\cos x)] dx\). Simplify the expression inside the integral before solving.

Step 4: Simplify and Compute the Integral

Simplify to \(A = \int_{0}^{2\pi} (2 - 2\cos x) dx\) and compute the integral. The integral evaluates to: \(A = [2x - 2\sin x]_{0}^{2\pi}\).

Step 5: Evaluate the Definite Integral

Substitute the limits \(0\) and \(2\pi\), and subtract to get \(A = (2*2\pi - 2\sin 2\pi) - (2*0 - 2\sin 0) = 4\pi\).

Key Concepts

Area between curves

To find the area between curves, first visualize the functions on a graph. The two functions given are:

• \(f(x) = \cos x\)
• \(g(x) = 2 - \cos x\)

These functions are plotted from the lower limit 0 to the upper limit \(2\pi\). Understanding the concept of the area between curves involves finding the space enclosed by these curves on this interval.
Simply put, you want to determine how much space lies between these two graphs. This happens by subtracting the value of the lower curve from the higher curve at any given point on the x-axis. In this case, the function \(g(x)\) is consistently above \(f(x)\) within the region we are considering.
This makes the integral easier to set up because we only have to take \(g(x) - f(x)\) into consideration. It is like stacking one graph on top of the other and finding the space left in between.

Definite integral

The definite integral is a powerful tool for calculating the area between two curves. A definite integral is represented by integral signs with limits at the top and bottom. The limits here are 0 and \(2\pi\).
We use the definite integral in problems like this to find an exact area and indicate a start and end point.

• Set up as: \(\int_{a}^{b} [\text{top function} - \text{bottom function}]\, dx\)

For this exercise, it is \(A = \int_{0}^{2\pi} [g(x) - f(x)]\, dx\).
When you solve this, you're summing up tiny slivers of area from left to right, from x = 0 to x = 2\(\pi\).
By performing this integration, you end up with a single numerical value representing the total area between the curves.

Trigonometric functions

Trigonometric functions like \(\cos(x)\) play a crucial role in this problem.
They are periodic functions, which means they repeat their values over regular intervals.
For \(\cos(x)\), the basic shape is known and it oscillates between -1 and 1.

• Meanwhile, \(g(x) = 2 - \cos(x)\) creates a shifted version of the \(\cos(x)\) graph upwards by 2 units.

This shift affects how the graphs relate to each other, demonstrating how changes in trigonometric functions influence the area between curves.
Understanding the properties of trigonometric functions helps you recognize how and why these functions interact the way they do on a graph.
When you are subtracting them inside the integral, these characteristics determine the result you get after performing the integration.

Step-by-step integration

Step-by-step integration is crucial for solving calculus problems, especially when finding areas between complex curves. Let's break this down:

• **Step 1**: Plot the curves. Visually understanding the problem is key. This comes from sketching \(f(x)\) and \(g(x)\) on a graph.
• **Step 2**: Set up the integral. This involves your basic formula for the area between curves: \(\int_{a}^{b} [\text{top curve} - \text{bottom curve}]\, dx\). Substitute the functions into this formula.
• **Step 3**: Simplify the expression. Before integrating, boil down the expressions: \(\int_{0}^{2\pi} (2 - 2\cos(x))\, dx\).
• **Step 4**: Compute the integral. This means performing the actual integration process, which here outputs: \( [2x - 2\sin(x)]_{0}^{2\pi} \).
• **Step 5**: Evaluate the limits. Substitute the limits into your integrated results to find the final area: \(4\pi\).

Linear thinking through the problem piece-by-piece illuminates how integration worlds practically for such tasks. By following a methodical pathway, you ensure accuracy in your calculations.