Question

The value of \(\int_{0}^{1} \frac{d x}{1+x}\) by Simpson's rule is (a) \(0.96815\) (b) \(0.63915\) (c) \(0.69315\) (d) 0,6 cho

Short answer

The answer is (c) 0.69315.

Step-by-step solution

Understanding Simpson's Rule

Simpson's Rule is a numerical method for approximating the value of a definite integral. It uses parabolic segments instead of straight lines to approximate the area under a curve. The formula for Simpson's Rule, when applying it to integrate a function \(f(x)\) over \([a, b]\), is: \[ \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{6} [f(a) + 4f((a+b)/2) + f(b)]. \] In this problem, \(f(x) = \frac{1}{1+x}\), \(a = 0\), and \(b = 1\).

Calculate Function Values at Endpoints and Midpoint

Firstly, calculate \(f(0)\), \(f(1)\), and \(f(0.5)\) for the given function \(f(x) = \frac{1}{1+x}\):- \(f(0) = \frac{1}{1+0} = 1\),- \(f(1) = \frac{1}{1+1} = \frac{1}{2}\),- \(f(0.5) = \frac{1}{1+0.5} = \frac{2}{3}.\)

Apply Simpson's Rule Formula

Substitute the calculated values into Simpson's rule formula. For \(a = 0\), \(b = 1\), \[ \int_{0}^{1} \frac{1}{1+x} \, dx \approx \frac{1-0}{6} [f(0) + 4f(0.5) + f(1)] = \frac{1}{6} [1 + 4 \times \frac{2}{3} + \frac{1}{2}]. \]

Simplify the Expression

Calculate the expression inside the brackets:- Calculate \(4 \times \frac{2}{3} = \frac{8}{3}\), so the expression becomes \(1 + \frac{8}{3} + \frac{1}{2}\).- Convert all terms to have a common denominator (6): \(1 = \frac{6}{6}\), \(\frac{8}{3} = \frac{16}{6}\), \(\frac{1}{2} = \frac{3}{6}\).- Add these up: \(\frac{6}{6} + \frac{16}{6} + \frac{3}{6} = \frac{25}{6}\).

Obtain the Final Result

Substitute the simplified result back into the overall formula: \[ \frac{1}{6} \times \frac{25}{6} = \frac{25}{36} \approx 0.69315. \]

Compare with Given Options

The closest option to the calculated integral value \(0.69315\) using Simpson's Rule is (c) \(0.69315\).

Key concepts

Numerical Integration

Numerical integration is a fundamental concept in calculus that focuses on computing the value of a definite integral without analytically solving it. This technique is particularly useful when dealing with complex integrals that are difficult or impossible to solve using elementary functions. Several methods of numerical integration exist, including trapezoidal rule, midpoint rule, and Simpson's rule. Each method approximates the area under a curve using different strategies, providing various levels of accuracy.

- **Trapezoidal Rule:** Uses line segments to approximate the area. - **Midpoint Rule:** Utilizes the midpoint of intervals. - **Simpson's Rule:** Employs parabolic segments for improved accuracy.
Simpson's Rule is known for its precision and is often preferred for many practical applications in engineering and the sciences.

Definite Integral

A definite integral is a fundamental concept in calculus representing the accumulated area under a curve within specified boundaries. It is denoted as \int_{a}^{b} f(x) \, dx\, where \(a\) and \(b\) are the lower and upper limits of integration. The function \(f(x)\) is the integrand.

The definite integral provides valuable information about total quantities, such as distance, area, and volume, depending on the context. Without numerical integration methods like Simpson's Rule, computing these integrals can be challenging, especially for functions that lack antiderivatives expressible in terms of elementary functions.

Understanding definite integrals is a cornerstone in calculus and serves as the gateway to solving many problems in physics, engineering, and beyond.

Approximation Methods

Approximation methods in mathematics are techniques used to estimate the values of mathematical expressions when exact computation is either not possible or too costly in terms of resources. These methods are crucial in various fields, including computational mathematics, engineering, and finance, where precision is important but sometimes hard to achieve.

- **Purpose:** To approximate the values of integrals, derivatives, or solutions to equations. - **Use Cases:** Helpful when dealing with complex equations or functions. - **Common Methods:** Taylor series, derivative approximations, and inverse interpolation.

In the context of integrating functions, Simpson's Rule is a widely used approximation method that increases accuracy by considering the curvature of the graph. This is a significant improvement over simpler methods like the trapezoidal rule, making Simpson's Rule a popular choice for approximating definite integrals.

Calculus Problem-Solving

Calculus problem-solving involves a structured approach to tackle problems related to differentiation and integration. Being adept at problem-solving in calculus means understanding the theoretical background and effectively applying various techniques and formulas to real-world scenarios.

- **Understanding Methods:** Like knowing when to use which numerical method, such as Simpson's Rule versus trapezoidal rule. - **Applying Techniques:** Using algebraic manipulation and numerical methods to simplify and solve equations. - **Analyzing Results:** Interpretation and comparison of results to check for accuracy or errors.

Solving calculus problems often requires a deep comprehension of mathematical concepts and the ability to connect them to practical applications. Practice, coupled with a sound grasp of methods like numerical integration, is key to mastering calculus problem-solving, enabling students and professionals to efficiently tackle and solve a wide range of mathematical challenges.