Problem 3 - Thomas Calculus

Question

The instructions for the integrals in Exercises \(1-10\) have two parts, one for the Trapezoidal Rule and one for Simpson's Rule. . $$ \int_{-1}^{1}\left(x^{2}+1\right) d x $$

Step-by-Step Solution

Answer

The integral is approximately 2 using the Trapezoidal Rule and \(\frac{8}{3}\) using Simpson's Rule.

Step 1: Understanding the Problem

We are tasked with approximating the integral \( \int_{-1}^{1}(x^2+1)\, dx \) using both the Trapezoidal Rule and Simpson's Rule. Since the integral limits are from \(-1\) to \(1\), we will perform calculations over this range.

Step 2: Apply the Trapezoidal Rule

The Trapezoidal Rule estimates the integral by dividing the area under the curve into trapezoids. The formula is: \[ T_n = \frac{b-a}{2n} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \] For simplicity, we'll use \(n=2\). Here, \(x_0=-1\), \(x_1=0\), \(x_2=1\). Then: \(T_2 = \frac{1 - (-1)}{2 \times 2} \left( f(-1) + 2f(0) + f(1) \right)\) \[ f(x) = x^2 + 1, \] \( f(-1) = 2, f(0) = 1, f(1) = 2 \). Substitute these, \( T_2 = \frac{1}{2} \times (2 + 2 \times 1 + 2) = 2 \).

Step 3: Apply Simpson's Rule

Simpson's Rule uses parabolic segments to approximate the area under the curve. The formula is: \[ S_n = \frac{b-a}{3n} \left( f(x_0) + 4\sum_{i=0}^{(n/2)-1} f(x_{2i+1}) + 2\sum_{i=1}^{(n/2)-1} f(x_{2i}) + f(x_n) \right). \] Using \(n=2\), \(S_2 = \frac{1 - (-1)}{3 \times 2} \left( f(-1) + 4f(0) + f(1) \right)\) and substitute: \(S_2 = \frac{1}{3} \times (2 + 4 \times 1 + 2) = \frac{8}{3}\).

Step 4: Comparing Results

The approximate integral value using the Trapezoidal Rule is 2, while using Simpson's Rule is \(\frac{8}{3}\). Simpson's Rule generally provides a more accurate approximation for polynomials of degree three or lower.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method for approximating definite integrals. It works by dividing the area under a curve into a series of trapezoids rather than perfect rectangles. This method is beneficial when you do not have a function that's easy to integrate analytically, or when you're looking for a quick approximate value.
To apply the Trapezoidal Rule, follow these steps:

• Divide the interval \([a, b]\) into \(n\) equal subintervals. The endpoints of these subintervals are the points at which you will evaluate your function.
• Calculate the area of each trapezoid using the formula for the trapezoidal approximation.
• Add up the areas of all the trapezoids to get the total approximate area under the curve.

The formula for the Trapezoidal Rule is:\[T_n = \frac{b-a}{2n} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)\] where \((x_0, x_1, ..., x_n)\) are the points within the interval. When using \( n = 2 \) for a simple example like \((x^2 + 1)\), you substitute into the formula for an approximate value, making it a quick way to estimate integrals.

Simpson's Rule

Simpson's Rule is another numerical integration technique, offering more accuracy for polynomial functions, especially those of degree three or less. It approximates the area under a curve by fitting parabolas to sections of the curve, rather than using straight lines like the Trapezoidal Rule.
Simpson's Rule is based on the idea of interpolation, using quadratic polynomials to achieve a better curve fit. Here's how it works:

• Choose even \( n \) subintervals for the interval \([a, b]\).
• Calculate function values at each of the points corresponding to these \( n+1 \) equally spaced intervals.
• Apply the Simpson’s formula, which is:\[S_n = \frac{b-a}{3n} \left( f(x_0) + 4\sum_{i=0}^{(n/2)-1} f(x_{2i+1}) + 2\sum_{i=1}^{(n/2)-1} f(x_{2i}) + f(x_n) \right)\]

The factor of 4 used in the Simpson's Rule formula applies extra weight to the midpoints of each subinterval. This generally results in higher precision for polynomials, demonstrating fewer errors as opposed to the trapezoidal method over the same number of subintervals.

Definite Integral

A definite integral is a fundamental concept in calculus, representing the accumulation of quantities or the total area under a curve within a specific interval, from \( a \) to \( b \). This is written as:\[\int_{a}^{b} f(x) \, dx\]
The definite integral of a function gives a number, unlike an indefinite integral that results in a general function. In practical terms, definite integrals are used to find net area, total displacement, and other accumulative measures.
In any problem involving definite integrals, it's important to:

• Identify the limits \( a \) and \( b \).
• Determine the function \( f(x) \) to be integrated.
• Use integration techniques or numerical methods, like Trapezoidal and Simpson's Rule, to find approximate values.

This concept is crucial in understanding the total behavior of functions over specific intervals, providing solutions to many real-world problems.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. These functions are of the form:\[f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\]
Polynomials are categorized by their degree, which is the highest power of the variable in the expression. For instance:

• A linear polynomial has degree 1, like \(x + 1\).
• A quadratic polynomial has degree 2, such as \(x^2 - 4x + 4\).
• A cubic polynomial has degree 3, with examples like \(2x^3 + x^2 + x - 5\).

Polynomial functions are continuous and differentiable, making them a common choice for approximations and analysis in mathematics. They are simple to manage and offer smooth curves that are ideal inputs for methods such as Trapezoidal and Simpson's Rule, due to their inherent differentiability and predictable behavior.