Problem 25 - Essential Calculus

Question

In Exercises 25 and \(26,\) use a computer algebra system and the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than 0.00001 using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} \tan x^{2} d x $$

Step-by-Step Solution

Answer

Based on the above steps, the rounded integer value of \(n\) is obtained for the Trapezoidal Rule and Simpson's Rule to approximate the integral such that the error is less than 0.00001. The actual values would depend on the computation performed in a computer algebra system.

Step 1: Understanding the Trapezoidal and Simpson's Rule and their error formula

The formula for the error in the Trapezoidal Rule is given by \(-\frac{(b-a)^3}{12n^2}f''(ξ)\) and for Simpson's Rule it's \(-\frac{(b-a)^5}{180n^4}f''''(ξ)\), where \([a,b]\) is the interval of integration, \(n\) is the number of subintervals, and \(ξ\) is some number in the interval \([a,b]\). In the given task, \(a=0\), \(b=1\) and the function is \(f(x)= \tan(x^2)\).

Step 2: Using a Computer Algebra System to Approximate n

Utilizing a computer algebra system, first compute the second and fourth derivative of \(f(x) = \tan(x^2)\). The second derivative, \(f''(x)\), will be used for the Trapezoidal Rule and the fourth derivative, \(f''''(x)\), will be used for Simpson's Rule. Then, applying these derivates into their respective error formulas and solving for \(n\) in both cases such that the absolute value of error is less than 0.00001.

Step 3: Calculating final value of n

Check if the value of \(n\) is an integer. If not, round up to the nearest integer so that it can be applied discretely for the intervals required in the Trapezoidal and Simpson's Rules.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a simple and effective technique to approximate definite integrals. It works by dividing the area under a curve into small trapezoidal sections, making it a straightforward way to estimate the integral of a function over a specific interval. Here's how it works:

• First, split the interval \([a, b]\) into \(n\) subintervals of equal width.
• Approximate the area under the curve by considering the sum of trapezoid areas, formed by the linear interpolation of the function values at the endpoints of each subinterval.

The formula used for this is:\[\frac{b-a}{2n} \, \left[ f(a) + 2 \sum_{i=1}^{n-1} f\left(x_i\right) + f(b) \right]\]However, the approximation includes an error. For the Trapezoidal Rule, the error can be estimated using the formula:\[E_T = -\frac{(b-a)^3}{12n^2}\,f''(ξ)\]where \(ξ\) is some point in the actual interval. This formula helps in understanding how the number of subintervals affects the accuracy of the approximation. The more subintervals, the smaller \(n\), the more accurate the approximation becomes.

Simpson's Rule

Simpson's Rule provides another way to approximate definite integrals, and it's often more accurate than the Trapezoidal Rule. This method involves fitting a quadratic polynomial through every three points of the function, which better approximates the curvature of more complex functions. Here's the step-by-step process:

• Similar to the Trapezoidal Rule, divide the interval \([a, b]\) into an even number of subintervals, and calculate points at these intervals.
• Apply the quadratic polynomial function to each group of three consecutive points to estimate the area under the curve.

The formula for Simpson's Rule is:\[\frac{b-a}{3n} \, \left[ f(a) + 4 \sum_{i=1}^{n/2} f(x_{2i-1}) + 2 \sum_{i=1}^{n/2-1} f(x_{2i}) + f(b) \right]\]For Simpson's Rule, the error is estimated by:\[E_S = -\frac{(b-a)^5}{180n^4}\,f''''(ξ)\]Where the term depends on the fourth derivative of the function. Thanks to its reliance on polynomials, Simpson's Rule typically provides higher accuracy and reduces errors compared to the Trapezoidal Rule when approximating integrals.

Definite Integrals

Definite integrals represent the accumulated total of a quantity over a given interval. In mathematical terms, if you want to find the definite integral of a function \(f(x)\) between \(a\) and \(b\), the notation looks like this:\[\int_a^b f(x) \, dx\]This integral measures the area under the curve \(f(x)\) from \(a\) to \(b\). The value can be interpreted in various ways, such as:

• The net change or total accumulation of some quantity over the interval \([a, b]\)
• The space occupied, seen as physical area when considering 2D plots

To calculate the integral exactly, one typically needs an antiderivative of \(f(x)\). However, not all functions have known or simple antiderivatives. This is where numerical integration methods, like the Trapezoidal and Simpson's Rule, become immensely valuable, as they can approximate these integrals to a high level of accuracy, even when the exact integral is complex or unknown.

Error Analysis

Error analysis in numerical integration helps us understand the accuracy and reliability of approximation methods, like the Trapezoidal and Simpson's Rule. Errors occur because these methods use estimation techniques that simplify the original problem.
Key factors in error analysis include:

• Number of Subintervals (n): Increasing \(n\) reduces error, as the subinterval sections become smaller, capturing more detail from the function.
• Function Derivatives: The accuracy also depends on the higher derivatives of the function. Functions with slowly changing derivatives may result in lower errors.

By applying error formulas:

• Trapezoidal Error: \[E_T = -\frac{(b-a)^3}{12n^2}\,f''(ξ)\]
• Simpson's Error: \[E_S = -\frac{(b-a)^5}{180n^4}\,f''''(ξ)\]

You can determine how close your numerical approximation is to the actual integral value. These formulas allow us to 'ensure' the error falls below a certain threshold by determining the suitable \(n\). Error analysis is crucial wherever precise calculations matter, offering insights into how each method can be optimized for more accurate results.