Problem 35 - Essential Calculus

Question

In Exercises \(35-40,\) sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=2 \sin x, \quad g(x)=\tan x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{3} $$

Step-by-Step Solution

Answer

The points of intersection of the graphs of \(f(x) = 2 \sin x\) and \(g(x) = \tan x\) are \(x = 0\) and \(x = \pi/4\). The total area of the region bounded by these two functions is given by the sum of the areas of two regions, defined from \(x=-\pi/3\) to \(x=0\) and \(x=0\) to \(x=\pi/3\), with \(f(x) = 2 \sin x\) as the top function and \(g(x) = \tan x\) as the bottom function in the integral.

Step 1: Plot Functions and Identify Intersections

Create a graph of the two functions \(f(x)=2 \sin x\) and \(g(x)=\tan x\) on the interval \(-\pi/3 \leq x \leq \pi/3\). Then, set \(2 \sin x = \tan x\) to determine the points where the graphs intersect. Simplify this to \(\sin x = \frac{\tan x}{2}\) and solve. The solutions are \(x = 0\), and \(x = \pi/4\).

Step 2: Determine Top and Bottom Functions

By analyzing the values of \(f(x)\) and \(g(x)\) at points in the interval \(-\pi/3 \leq x \leq \pi/3\), determine that \(f(x)\) is the top function and \(g(x)\) is the bottom function.

Step 3: Calculate Areas and Sum

Calculate the areas of the regions separately using the formula for the area between curves: \(A = \int_{a}^{b} [f(x)-g(x)] dx\). This results in two areas, where the first region is defined from \(x=-\pi/3\) to \(x=0\) and the second region is defined from \(x=0\) to \(x=\pi/3\). The total area enclosed by the curves is the sum of these areas.

Key Concepts

Integrating Trigonometric Functions

When calculating the area between curves involving trigonometric functions like sine and tangent, we utilize the principles of integration.
For instance, finding the area under the curve of a function like \(f(x) = 2 \sin x\) and above a function like \(g(x) = \tan x\) requires subtracting one function from the other within an integral. The process of integrating trigonometric functions often involves using identities to simplify the integrand before applying the fundamental theorem of calculus.

For example, when integrating \( \sin x\), a common antiderivative is \( -\cos x\), taking into account the constant of integration when the bounds are indefinite. If the functions involved are more complex or products of trigonometric functions, one might need to use specific methods such as trigonometric substitution or integration by parts. In this case, no such complexities arise; however, knowing these techniques can be crucial for more challenging problems.

Area of Bounded Region

The area of a bounded region between two curves is found by taking the definite integral of the top function minus the bottom function over the interval of interest.
It is vital to first identify the correct interval, as well as which function is the 'top' (the one with the higher output values) and which is the 'bottom' (the one with the lower output values). This ensures you subtract the functions in the correct order.

For the functions given, \(f(x) = 2 \sin x\) and \(g(x) = \tan x\), we subtract the bottom function from the top function within a definite integral. In the exercise, the area is calculated in two parts since the bounds include an interval where the curves intersect, resulting in two distinct regions. The definite integrals for each region are calculated and then added together to find the total area. In mathematical terms, the area \(A\) is given by the sum of the integrals \( \int_{a}^{c} [f(x) - g(x)] \,dx\) and \( \int_{c}^{b} [f(x) - g(x)] \,dx\), where \(a\), \(b\), and \(c\) represent the intersection points and bounds of integration.

Sketching Graphs of Functions

Sketching the graph of a function is a fundamental skill in calculus. It helps you visualize the behavior of the function and identify key features such as intercepts, maxima, minima, and points of intersection between the curves.
To sketch trigonometric functions, like those in our exercise, \(f(x) = 2 \sin x\) and \(g(x) = \tan x\), it's beneficial to understand their periodicity, amplitude, and phase shift.

Key Steps in Sketching Graphs

• Plot key points: Identify points where the function crosses the x-axis (x-intercepts), the y-axis (y-intercepts), and any maxima or minima within the interval.
• Consider asymptotes: For functions like \(\tan x\), which have vertical asymptotes, be sure to draw these lines to represent the points where the function approaches infinity.
• Identify the interval: In this case, we are considering the interval \( -\frac{\pi}{3} \leq x \leq \frac{\pi}{3} \).
• Analyze symmetry and periodicity: Use the periodic nature of trigonometric functions to replicate the pattern within the given range.

In our exercise, sketching the graphs precisely allows for easier identification of the intersection points and visualization of the area that needs to be calculated.