Approximating the Integral

Oct 13, 2025 by windailys.com 127 views

Introduction

In mathematics, definite integrals are used to calculate the area under curves and are a fundamental concept in calculus. However, calculating definite integrals analytically can be challenging, especially for complex functions. In such cases, numerical methods like the Trapezoidal rule can be employed to approximate the integral. In this article, we will explore the Trapezoidal rule and its application to approximate the definite integral ${$ \int_6^9 \frac{3}{(s-5)^2} , ds$}$.

The Trapezoidal Rule

The Trapezoidal rule is a numerical method used to approximate the value of a definite integral. It works by dividing the area under the curve into small trapezoids and summing the areas of these trapezoids. The Trapezoidal rule is based on the formula:

${$ \int_a^b f(x) , dx \approx \frac{h}{2} \left[ f(a) + 2f(a+h) + 2f(a+2h) + \cdots + 2f(b-h) + f(b) \right]$}$

where ${$ h = \frac{b-a}{n}$] is the width of each subinterval, and [ $n\$}$ is the number of subintervals.

Error Analysis

The error in the Trapezoidal rule is given by the formula:

${$ E = -\frac{b-a}{12} h^2 f''(\xi)$}$

where ${$ \xi$}$ is a point in the interval ${$ [a,b]$]. The error is proportional to the square of the width of each subinterval, [ $h^2\$. To ensure that the error is less than a certain magnitude, \[$ \square$}$, we need to choose a sufficiently large value of ${$ n$. The number of subintervals, [ $n\$, is related to the width of each subinterval, \[$ h$, by the formula:

[ $h = \frac{b-a}{n}\$}$

Approximating the Integral

To approximate the integral ${$ \int_6^9 \frac{3}{(s-5)^2} , ds$, we need to choose a value of [ $n\$. To ensure that the error is less than \[$ \square$, we can use the formula:

[ $n \geq \frac{12(b-a)}{h^2 \left| f''(\xi) \right|}\$}$

where ${$ f''(\xi)$ is the second derivative of the function [$f(s) = \frac{3}{(s-5)^2}$.

Calculating the Second Derivative

To calculate the second derivative of the function [$f(s) = \frac{3}{(s-5)^2}$, we can use the power rule of differentiation:

[ $f'(s) = -\frac{6}{(s-5)^3}\$}$

Differentiating again, we get:

${$ f''(s) = \frac{18}{(s-5)^4}$}$

**Choosing the Value of ${$ n$

To choose the value of [ $n\$, we need to calculate the second derivative of the function at a point in the interval \[$ [6,9]$. Let's choose [$\xi = 7.5$. Then, the second derivative is:

[ $f''(7.5) = \frac{18}{(7.5-5)^4} = \frac{18}{2.5^4} = \frac{18}{39.0625} = 0.4607\$}$

Now, we can use the formula:

${$ n \geq \frac{12(b-a)}{h^2 \left| f''(\xi) \right|}$

to choose the value of [$n$. Plugging in the values, we get:

[$n \geq \frac{12(9-6)}{h^2 \left| 0.4607 \right|}$

To ensure that the error is less than [ $\square\$, we need to choose a sufficiently large value of \[$ n$. Let's choose [ $h = 0.1\$. Then, the value of \[$ n$ is:

[$n \geq \frac{12(9-6)}{(0.1)^2 \left| 0.4607 \right|} = \frac{12(3)}{0.01 \cdot 0.4607} = \frac{36}{0.004607} = 7831.5$

$Since \[$n\$}$ must be an integer, we need to choose the smallest integer greater than or equal to ${$7831.5\$. Therefore, the value of \[$n\$}$ must be at least ${$6364\$.<br/> **Frequently Asked Questions (FAQs)**$

Q: What is the Trapezoidal rule?

A: The Trapezoidal rule is a numerical method used to approximate the value of a definite integral. It works by dividing the area under the curve into small trapezoids and summing the areas of these trapezoids.

Q: What is the formula for the Trapezoidal rule?

A: The formula for the Trapezoidal rule is:

[$\int_a^b f(x) , dx \approx \frac{h}{2} \left[ f(a) + 2f(a+h) + 2f(a+2h) + \cdots + 2f(b-h) + f(b) \right]$

where [ $h = \frac{b-a}{n}\$] is the width of each subinterval, and \[$ n$}$ is the number of subintervals.

Q: What is the error in the Trapezoidal rule?

A: The error in the Trapezoidal rule is given by the formula:

${$ E = -\frac{b-a}{12} h^2 f''(\xi)$}$

where ${$ \xi$}$ is a point in the interval ${$ [a,b]$]. The error is proportional to the square of the width of each subinterval, [$h^2$.

Q: How do I choose the value of [ $n}$ ?

A: To choose the value of ${$ n$, you need to calculate the second derivative of the function at a point in the interval [$[a,b]$. Then, you can use the formula:

[$n \geq \frac{12(b-a)}{h^2 \left| f''(\xi) \right|}$

to choose the value of [ $n\$. The value of \[$ n$}$ must be an integer, so you need to choose the smallest integer greater than or equal to the calculated value.

Q: What is the second derivative of the function [$f(s) = \frac{3}{(s-5)^2}$?

A: The second derivative of the function [$f(s) = \frac{3}{(s-5)^2}$ is:

[$f''(s) = \frac{18}{(s-5)^4}$

Q: How do I calculate the second derivative of a function?

A: To calculate the second derivative of a function, you need to differentiate the function twice. The power rule of differentiation states that if [ $f(x) = x^n\$, then \[$ f'(x) = nx^{n-1}$. You can use this rule to differentiate the function and then differentiate the result again to get the second derivative.

Q: What is the significance of the Trapezoidal rule?

A: The Trapezoidal rule is a numerical method used to approximate the value of a definite integral. It is useful when the function is difficult to integrate analytically or when the integral is too complex to evaluate exactly. The Trapezoidal rule is also used in numerical analysis and computer science to approximate the value of definite integrals.

Q: What are some common applications of the Trapezoidal rule?

A: The Trapezoidal rule is used in a variety of applications, including:

Numerical integration
Approximating the value of definite integrals
Solving differential equations
Computer graphics
Signal processing

Q: How do I implement the Trapezoidal rule in a programming language?

A: To implement the Trapezoidal rule in a programming language, you need to write a function that takes the function, the limits of integration, and the number of subintervals as input. The function should then calculate the width of each subinterval, calculate the function values at each point, and sum the areas of the trapezoids to get the approximate value of the integral.