Applying the Rules to the Given Integral

Oct 13, 2025 by windailys.com 76 views

Introduction

In mathematics, approximating the value of a definite integral is a crucial task, especially when the function is complex or cannot be integrated analytically. The Trapezoidal Rule, Midpoint Rule, and Simpson's Rule are three popular methods used to approximate the value of a definite integral. In this article, we will explore these methods and apply them to approximate the integral $\int_1^3 \frac{\ln(x)}{3+x} \, dx$ with $n=10$ .

The Trapezoidal Rule

The Trapezoidal Rule is a method for approximating the value of a definite integral by dividing the area under the curve into trapezoids and summing the areas of the trapezoids. The formula for the Trapezoidal Rule is:

T_n = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]

where $h$ is the width of each subinterval, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the point $x_i$ .

The Midpoint Rule

The Midpoint Rule is another method for approximating the value of a definite integral by dividing the area under the curve into rectangles and summing the areas of the rectangles. The formula for the Midpoint Rule is:

M_n = h \left[ f(x_0) + f(x_1) + f(x_2) + \cdots + f(x_{n-1}) + f(x_n) \right]

where $h$ is the width of each subinterval, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the midpoint of the subinterval.

Simpson's Rule

Simpson's Rule is a method for approximating the value of a definite integral by dividing the area under the curve into parabolic segments and summing the areas of the parabolic segments. The formula for Simpson's Rule is:

S_n = \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, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the point $x_i$ .

Applying the Rules to the Given Integral

To apply the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule to the given integral, we need to divide the interval $[1,3]$ into $n=10$ subintervals, each of width $h=\frac{3-1}{10}=\frac{1}{5}$ . We then evaluate the function $\frac{\ln(x)}{3+x}$ at the points $x_0=1$ , $x_1=1.1$ , $x_2=1.2$ , $x_3=1.3$ , $x_4=1.4$ , $x_5=1.5$ , $x_6=1.6$ , $x_7=1.7$ , $x_8=1.8$ , $x_9=1.9$ , and $x_{10}=3$ .

Calculating the Trapezoidal Rule

Using the Trapezoidal Rule formula, we get:

T_{10} = \frac{1/5}{2} \left[ \frac{\ln(1)}{3+1} + 2\frac{\ln(1.1)}{3+1.1} + 2\frac{\ln(1.2)}{3+1.2} + 2\frac{\ln(1.3)}{3+1.3} + 2\frac{\ln(1.4)}{3+1.4} + 2\frac{\ln(1.5)}{3+1.5} + 2\frac{\ln(1.6)}{3+1.6} + 2\frac{\ln(1.7)}{3+1.7} + 2\frac{\ln(1.8)}{3+1.8} + 2\frac{\ln(1.9)}{3+1.9} + \frac{\ln(3)}{3+3} \right]

Evaluating the expression, we get:

T_{10} = 0.2652

Calculating the Midpoint Rule

Using the Midpoint Rule formula, we get:

M_{10} = \frac{1}{5} \left[ \frac{\ln(1)}{3+1} + \frac{\ln(1.05)}{3+1.05} + \frac{\ln(1.15)}{3+1.15} + \frac{\ln(1.25)}{3+1.25} + \frac{\ln(1.35)}{3+1.35} + \frac{\ln(1.45)}{3+1.45} + \frac{\ln(1.55)}{3+1.55} + \frac{\ln(1.65)}{3+1.65} + \frac{\ln(1.75)}{3+1.75} + \frac{\ln(1.85)}{3+1.85} + \frac{\ln(3)}{3+3} \right]

Evaluating the expression, we get:

M_{10} = 0.2484

Calculating Simpson's Rule

Using Simpson's Rule formula, we get:

S_{10} = \frac{1/5}{3} \left[ \frac{\ln(1)}{3+1} + 4\frac{\ln(1.1)}{3+1.1} + 2\frac{\ln(1.2)}{3+1.2} + 4\frac{\ln(1.3)}{3+1.3} + 2\frac{\ln(1.4)}{3+1.4} + 4\frac{\ln(1.5)}{3+1.5} + 2\frac{\ln(1.6)}{3+1.6} + 4\frac{\ln(1.7)}{3+1.7} + 2\frac{\ln(1.8)}{3+1.8} + 4\frac{\ln(1.9)}{3+1.9} + \frac{\ln(3)}{3+3} \right]

Evaluating the expression, we get:

S_{10} = 0.2621

Discussion

In this article, we have applied the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule to approximate the integral $\int_1^3 \frac{\ln(x)}{3+x} \, dx$ with $n=10$ . The results are:

$T_{10} = 0.2652$
$M_{10} = 0.2484$
$S_{10} = 0.2621$

The Trapezoidal Rule and Simpson's Rule give similar results, while the Midpoint Rule gives a slightly lower value. This is because the Midpoint Rule uses the function values at the midpoints of the subintervals, while the Trapezoidal Rule and Simpson's Rule use the function values at the endpoints and midpoints, respectively.

In conclusion, the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule are all useful methods for approximating the value of a definite integral. The choice of method depends on the specific problem and the desired level of accuracy.

Introduction

In our previous article, we explored the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule as methods for approximating the value of a definite integral. In this article, we will answer some frequently asked questions about these methods and provide additional insights into their application.

Q: What is the Trapezoidal Rule?

A: The Trapezoidal Rule is a method for approximating the value of a definite integral by dividing the area under the curve into trapezoids and summing the areas of the trapezoids. The formula for the Trapezoidal Rule is:

T_n = \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right]

where $h$ is the width of each subinterval, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the point $x_i$ .

Q: What is the Midpoint Rule?

A: The Midpoint Rule is another method for approximating the value of a definite integral by dividing the area under the curve into rectangles and summing the areas of the rectangles. The formula for the Midpoint Rule is:

M_n = h \left[ f(x_0) + f(x_1) + f(x_2) + \cdots + f(x_{n-1}) + f(x_n) \right]

where $h$ is the width of each subinterval, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the midpoint of the subinterval.

Q: What is Simpson's Rule?

A: Simpson's Rule is a method for approximating the value of a definite integral by dividing the area under the curve into parabolic segments and summing the areas of the parabolic segments. The formula for Simpson's Rule is:

S_n = \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, $x_0$ and $x_n$ are the endpoints of the interval, and $f(x_i)$ is the value of the function at the point $x_i$ .

Q: How do I choose between the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule?

A: The choice of method depends on the specific problem and the desired level of accuracy. The Trapezoidal Rule and Simpson's Rule are generally more accurate than the Midpoint Rule, but they require more function evaluations. The Midpoint Rule is simpler to implement and requires fewer function evaluations, but it may not be as accurate.

Q: What is the relationship between the Trapezoidal Rule, Midpoint Rule, and Simpson's Rule?

A: The Trapezoidal Rule and Simpson's Rule are both based on the idea of approximating the area under the curve by dividing it into smaller segments. The Trapezoidal Rule uses trapezoids, while Simpson's Rule uses parabolic segments. The Midpoint Rule is a simpler method that uses rectangles to approximate the area under the curve.

Q: Can I use these methods for approximating the value of a definite integral with a non-rectangular interval?

A: Yes, you can use these methods for approximating the value of a definite integral with a non-rectangular interval. However, you will need to divide the interval into smaller subintervals and apply the method to each subinterval separately.

Q: How do I implement these methods in a programming language?

A: You can implement these methods in a programming language such as Python, MATLAB, or C++. The implementation will depend on the specific language and the desired level of accuracy.

Q: What are some common applications of these methods?

A: These methods are commonly used in a variety of fields, including physics, engineering, economics, and computer science. They are used to approximate the value of definite integrals that arise in the solution of differential equations, optimization problems, and other mathematical models.

Q: What are some common pitfalls to avoid when using these methods?

A: Some common pitfalls to avoid when using these methods include:

Using too few subintervals, which can lead to inaccurate results
Using too many subintervals, which can lead to unnecessary computational expense
Failing to check for convergence, which can lead to inaccurate results
Failing to use a sufficient number of decimal places, which can lead to inaccurate results

By understanding the strengths and weaknesses of these methods and avoiding common pitfalls, you can use them effectively to approximate the value of definite integrals in a variety of applications.