Use Simpson's Rule With $n=4$ To Approximate$\int_1^3 \frac{\cos (x)}{x} \, Dx$.Keep At Least 2 Decimal Places Of Accuracy In Your Final Answer.

Introduction

Simpson's rule is a powerful numerical method for approximating definite integrals. It is a refinement of the trapezoidal rule and is particularly useful for approximating integrals that have a high degree of smoothness. In this article, we will use Simpson's rule with $n=4$ to approximate the definite integral $\int_1^3 \frac{\cos (x)}{x} \, dx$. We will also discuss the accuracy of the approximation and provide a final answer with at least 2 decimal places of accuracy.

Simpson's Rule

Simpson's rule is a numerical method for approximating definite integrals. It is based on the idea of approximating the area under a curve by dividing it into small parabolic segments. The rule states that if $f(x)$ is a function that is continuous on the interval $[a,b]$, then the definite integral $\int_a^b f(x) \, dx$ can be approximated by the formula:

$$\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]$$

where $h$ is the width of each subinterval, $x_0, x_1, x_2, \ldots, x_n$ are the points of subdivision, and $f(x_i)$ is the value of the function at each point.

Approximating the Definite Integral

To approximate the definite integral $\int_1^3 \frac{\cos (x)}{x} \, dx$ using Simpson's rule, we need to divide the interval $[1,3]$ into $n=4$ subintervals. The points of subdivision are $x_0=1$, $x_1=1.5$, $x_2=2$, $x_3=2.5$, and $x_4=3$.

We can now use Simpson's rule to approximate the definite integral:

$$\int_1^3 \frac{\cos (x)}{x} \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4) \right]$$

where $h=1$ is the width of each subinterval, and $f(x_i)=\frac{\cos (x_i)}{x_i}$ is the value of the function at each point.

Calculating the Approximation

To calculate the approximation, we need to evaluate the function at each point:

$$f(x_0)=\frac{\cos (1)}{1} \approx 0.5403$$

$$f(x_1)=\frac{\cos (1.5)}{1.5} \approx 0.3849$$

$$f(x_2)=\frac{\cos (2)}{2} \approx 0.2454$$

$$f(x_3)=\frac{\cos (2.5)}{2.5} \approx 0.1411$$

$$f(x_4)=\frac{\cos (3)}{3} \approx 0.0872$$

We can now substitute these values into the formula:

$$\int_1^3 \frac{\cos (x)}{x} \, dx \approx \frac{1}{3} \left[ 0.5403 + 4(0.3849) + 2(0.2454) + 4(0.1411) + 0.0872 \right]$$

$$\int_1^3 \frac{\cos (x)}{x} \, dx \approx \frac{1}{3} \left[ 0.5403 + 1.5396 + 0.4908 + 0.5644 + 0.0872 \right]$$

$$\int_1^3 \frac{\cos (x)}{x} \, dx \approx \frac{1}{3} \left[ 3.1223 \right]$$

$$\int_1^3 \frac{\cos (x)}{x} \, dx \approx 1.0408$$

Conclusion

In this article, we used Simpson's rule with $n=4$ to approximate the definite integral $\int_1^3 \frac{\cos (x)}{x} \, dx$. We calculated the approximation using the formula and obtained a value of approximately $1.0408$. This value has at least 2 decimal places of accuracy, as required.

Code

Here is the Python code to calculate the approximation:

import numpy as np

def f(x):
    return np.cos(x) / x

def simpson_rule(f, a, b, n):
    h = (b - a) / n
    x = np.linspace(a, b, n + 1)
    y = f(x)
    return h / 3 * (y[0] + 4 * y[1] + 2 * y[2] + 4 * y[3] + y[4])

a = 1
b = 3
n = 4

result = simpson_rule(f, a, b, n)
print(result)

Q: What is Simpson's rule?

A: Simpson's rule is a numerical method for approximating definite integrals. It is a refinement of the trapezoidal rule and is particularly useful for approximating integrals that have a high degree of smoothness.

Q: How does Simpson's rule work?

A: Simpson's rule works by dividing the interval of integration into small parabolic segments. The rule states that if $f(x)$ is a function that is continuous on the interval $[a,b]$, then the definite integral $\int_a^b f(x) \, dx$ can be approximated by the formula:

$$\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]$$

where $h$ is the width of each subinterval, $x_0, x_1, x_2, \ldots, x_n$ are the points of subdivision, and $f(x_i)$ is the value of the function at each point.

Q: What are the advantages of Simpson's rule?

A: The advantages of Simpson's rule include:

• It is more accurate than the trapezoidal rule for smooth functions.
• It is particularly useful for approximating integrals that have a high degree of smoothness.
• It is easy to implement and requires minimal computational resources.

Q: What are the disadvantages of Simpson's rule?

A: The disadvantages of Simpson's rule include:

• It requires the function to be continuous on the interval of integration.
• It requires the function to be smooth, meaning that it has a high degree of continuity.
• It can be less accurate than other numerical methods for certain types of functions.

Q: How do I choose the number of subintervals (n) for Simpson's rule?

A: The choice of the number of subintervals (n) depends on the desired accuracy of the approximation. A larger value of n will result in a more accurate approximation, but will also require more computational resources.

Q: Can I use Simpson's rule for approximating integrals with singularities?

A: No, Simpson's rule is not suitable for approximating integrals with singularities. The rule requires the function to be continuous on the interval of integration, which is not the case for functions with singularities.

Q: Can I use Simpson's rule for approximating integrals with discontinuities?

A: No, Simpson's rule is not suitable for approximating integrals with discontinuities. The rule requires the function to be continuous on the interval of integration, which is not the case for functions with discontinuities.

Q: How do I implement Simpson's rule in a programming language?

A: Implementing Simpson's rule in a programming language involves defining the function to be integrated, dividing the interval of integration into small subintervals, and applying the Simpson's rule formula to approximate the integral.

Code

Here is an example of how to implement Simpson's rule in Python:

import numpy as np

def f(x):
    return np.cos(x) / x

def simpson_rule(f, a, b, n):
    h = (b - a) / n
    x = np.linspace(a, b, n + 1)
    y = f(x)
    return h / 3 * (y[0] + 4 * y[1] + 2 * y[2] + 4 * y[3] + y[4])

a = 1
b = 3
n = 4

result = simpson_rule(f, a, b, n)
print(result)

This code defines the function f(x) and the Simpson's rule function simpson_rule(f, a, b, n). It then calculates the approximation using the formula and prints the result.