Problem 36 - Essential Calculus

Question

Sketch the region bounded by the graphs of the functions, and find the area of the region. $$ f(x)=\sin x, g(x)=\cos 2 x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{6} $$

Step-by-Step Solution

Answer

The area of the region bounded by the two given functions is equal to the absolute value of the integral of the difference of these two functions, evaluated from the lower to the upper limit that we find as intersection points in step 2.

Step 1: Plotting the given functions

First plot the function \(f(x) = \sin x\) in the interval \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{6}\). Then plot \(g(x) = \cos 2x\) in the same interval. We need to identify the region bounded by the two graphs. This region would be the area that these functions share.

Step 2: Find the intersection points of the two graphs

Set the two functions equal to each other and solve for \(x\) in order to find the \(x\) coordinates at which the graphs intersect. Thus, we have to solve: \[\sin x = \cos 2x\] Once we have the intersection points, we can label these points on our graph.

Step 3: Calculate the area bounded by the two graphs

To calculate the area bounded by the two graphs, we can use the concept of integration. The area \(A\) bounded by the graphs over the interval \(\[a, b\]\) is given by \[\int_{a}^{b} |f(x) - g(x)| dx\]. Plug in the functions given and the bounds we have found in the previous step. Since we can't predict whether \(sinx\) or \(cos2x\) is greater in the given range, we will take absolute of the entire difference. Evaluating the integral will give the total area of the region bounded by the two functions, \(f(x)\) and \(g(x)\).

Key Concepts

Integration

Understanding the concept of integration is crucial when attempting to find the area between curves in calculus. Integration is essentially a way to measure how much space is underneath a curve on a graph. If you imagine a curve as a hilly landscape, integration would allow you to calculate the total area of land from one hill to another.

In the context of the given problem, integration is used to find the area between the functions given by \( f(x) = \text{sin}\,x \) and \( g(x) = \text{cos}\,2x \) over the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{6} \). The integral requires the absolute difference of two functions, which ensures you count all the space between the curves, no matter which function is on top. Therefore, the integral to compute the area is \( \int_{a}^{b} |f(x) - g(x)| \,dx \), where \( a \) and \( b \) are the points where the curves intersect, or the boundaries of the interval provided.

Intersection Points

Intersection points between two or more functions are the X-coordinates where their corresponding Y-values are equal. In graphical terms, they're the exact points where the different function curves meet or cross each other. When calculating the area between curves, it is essential to accurately determine where these intersection points are because they often serve as the limits of integration.

To find these points for the functions \( f(x) = \text{sin}\,x \) and \( g(x) = \text{cos}\,2x \), we set them equal to each other and solve for \( x \). The solutions to the equation \( \text{sin}\,x = \text{cos}\,2x \) will give us the required points. Finding intersection points accurately ensures the area calculation is bounded correctly, and it is also important for graphically representing the region in question.

Trigonometric Functions

Trigonometric functions are fundamental in calculus, especially when dealing with periodic phenomena. Functions like \( \text{sin}\,x \) and \( \text{cos}\,x \) map the circular motion onto a graph, creating waves that represent repetitive patterns. These functions are the bread and butter of many calculus problems, including ones that involve finding areas between curves.

The given problem involves \( f(x) = \text{sin}\,x \) and \( g(x) = \text{cos}\,2x \), which are trigonometric functions with different periods and amplitudes. Understanding their shapes and properties is necessary to anticipate where intersection points might occur as well as the nature of the area bounded by them. Learning how to manipulate these functions algebraically and graphically is key in solving calculus problems dealing with areas between curves.