Finding the Area Bounded by the Graph and the Axes

Oct 13, 2025 by miodeya.com 104 views

===========================================================

Introduction

In mathematics, finding the area bounded by a graph and the x- and y-axes is a fundamental problem that has numerous applications in various fields. In this article, we will focus on finding the area bounded by the graph of $$y=\cos \left(x+\frac{\pi}{4}\right)$$, the x- and y-axes, and the line $$x=\pi$$. This problem involves the use of trigonometric functions, particularly the cosine function, and requires a deep understanding of the properties of these functions.

Understanding the Graph

The given graph is a cosine function with a phase shift of $$\frac{\pi}{4}$$. This means that the graph is shifted to the left by $$\frac{\pi}{4}$$ units. The graph is also reflected about the y-axis, which means that the graph is symmetric about the y-axis.

To understand the graph, we need to consider the following:

The graph is periodic with a period of $$2\pi$$.
The graph has a maximum value of 1 and a minimum value of -1.
The graph is symmetric about the y-axis.

Finding the Area Bounded by the Graph and the Axes

To find the area bounded by the graph and the axes, we need to integrate the function with respect to x from 0 to $$\pi$$. This will give us the area under the graph between these two limits.

The area under the graph can be calculated using the following integral:

\int_{0}^{\pi} \cos \left(x+\frac{\pi}{4}\right) dx

Evaluating the Integral

To evaluate the integral, we need to use the following trigonometric identity:

\cos (A+B) = \cos A \cos B - \sin A \sin B

Using this identity, we can rewrite the integral as:

\int_{0}^{\pi} \cos \left(x+\frac{\pi}{4}\right) dx = \int_{0}^{\pi} \left(\cos x \cos \frac{\pi}{4} - \sin x \sin \frac{\pi}{4}\right) dx

Solving the Integral

To solve the integral, we need to integrate each term separately.

The first term can be integrated as follows:

\int_{0}^{\pi} \cos x \cos \frac{\pi}{4} dx = \cos \frac{\pi}{4} \int_{0}^{\pi} \cos x dx

The second term can be integrated as follows:

\int_{0}^{\pi} \sin x \sin \frac{\pi}{4} dx = \sin \frac{\pi}{4} \int_{0}^{\pi} \sin x dx

Evaluating the Integrals

To evaluate the integrals, we need to use the following trigonometric identities:

$\int \cos x dx = \sin x + C$
$\int \sin x dx = -\cos x + C$

Using these identities, we can evaluate the integrals as follows:

\int_{0}^{\pi} \cos x dx = \sin x \bigg|_{0}^{\pi} = \sin \pi - \sin 0 = 0 - 0 = 0

\int_{0}^{\pi} \sin x dx = -\cos x \bigg|_{0}^{\pi} = -\cos \pi + \cos 0 = -(-1) + 1 = 2

Finding the Area

To find the area, we need to substitute the values of the integrals back into the original equation.

The area under the graph can be calculated as follows:

\int_{0}^{\pi} \cos \left(x+\frac{\pi}{4}\right) dx = \cos \frac{\pi}{4} \int_{0}^{\pi} \cos x dx - \sin \frac{\pi}{4} \int_{0}^{\pi} \sin x dx

= \cos \frac{\pi}{4} (0) - \sin \frac{\pi}{4} (2)

= 0 - \frac{\sqrt{2}}{2} (2)

= -\sqrt{2}

However, since we are finding the area under the graph, we need to take the absolute value of the result.

The area under the graph is:

\left|-\sqrt{2}\right| = \sqrt{2}

Conclusion

In this article, we have found the area bounded by the graph of $$y=\cos \left(x+\frac{\pi}{4}\right)$$, the x- and y-axes, and the line $$x=\pi$$. The area under the graph is $$\sqrt{2}$$ square units.

This problem involves the use of trigonometric functions, particularly the cosine function, and requires a deep understanding of the properties of these functions. The graph is periodic with a period of $$2\pi$$, has a maximum value of 1 and a minimum value of -1, and is symmetric about the y-axis.

The area under the graph can be calculated using the following integral:

\int_{0}^{\pi} \cos \left(x+\frac{\pi}{4}\right) dx

The integral can be evaluated using the following trigonometric identity:

\cos (A+B) = \cos A \cos B - \sin A \sin B

The area under the graph is $$\sqrt{2}$$ square units.

References

[1] Calculus by Michael Spivak
[2] Trigonometry by I.M. Gelfand
[3] Mathematics for Computer Science by Eric Lehman

Future Work

In the future, we can explore other problems involving trigonometric functions and their applications in various fields. We can also investigate other methods for finding the area under a graph, such as using numerical methods or approximations.

Code

The following code can be used to calculate the area under the graph:

import numpy as np
from scipy.integrate import quad
def integrand(x):
return np.cos(x + np.pi/4)
result, error = quad(integrand, 0, np.pi)
print("The area under the graph is:", result)

This code uses the quad function from the scipy.integrate module to numerically integrate the function and calculate the area under the graph.

===========================================================

Introduction

In our previous article, we discussed how to find the area bounded by the graph of $$y=\cos \left(x+\frac{\pi}{4}\right)$$, the x- and y-axes, and the line $$x=\pi$$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the period of the graph?

A: The period of the graph is $$2\pi$$.

Q: What is the maximum value of the graph?

A: The maximum value of the graph is 1.

Q: What is the minimum value of the graph?

A: The minimum value of the graph is -1.

Q: How do I find the area under the graph?

A: To find the area under the graph, you need to integrate the function with respect to x from 0 to $$\pi$$. This will give you the area under the graph between these two limits.

Q: What is the formula for finding the area under the graph?

A: The formula for finding the area under the graph is:

\int_{0}^{\pi} \cos \left(x+\frac{\pi}{4}\right) dx

Q: How do I evaluate the integral?

A: To evaluate the integral, you need to use the following trigonometric identity:

\cos (A+B) = \cos A \cos B - \sin A \sin B

Q: What is the final answer for the area under the graph?

A: The final answer for the area under the graph is $$\sqrt{2}$$ square units.

Q: Can I use numerical methods to find the area under the graph?

A: Yes, you can use numerical methods to find the area under the graph. The following code can be used to calculate the area under the graph:

import numpy as np
from scipy.integrate import quad
def integrand(x):
return np.cos(x + np.pi/4)
result, error = quad(integrand, 0, np.pi)
print("The area under the graph is:", result)

Q: What are some real-world applications of finding the area under a graph?

A: Finding the area under a graph has numerous real-world applications in various fields, including:

Physics: Finding the area under a graph can help you calculate the work done by a force on an object.
Engineering: Finding the area under a graph can help you calculate the stress on a material.
Economics: Finding the area under a graph can help you calculate the total revenue of a company.

Q: Can I use this method to find the area under any graph?

A: No, this method can only be used to find the area under a graph that is a function of x. If the graph is a function of y, you will need to use a different method.

Q: What are some common mistakes to avoid when finding the area under a graph?

A: Some common mistakes to avoid when finding the area under a graph include:

Not using the correct limits of integration.
Not using the correct formula for the area under the graph.
Not evaluating the integral correctly.