Problem 17 - Essential Calculus

Question

In Exercises \(17-20,\) use the error formulas in Theorem 4.19 to estimate the error in approximating the integral, with \(n=4\), using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{2} x^{3} d x $$

Step-by-Step Solution

Answer

The error in approximating the integral using both the Trapezoidal Rule and Simpson's Rule is 0.

Step 1: Applying the Trapezoidal Rule

We first need to identify the fourth derivative of the function \(x^3\) because the error formula for the trapezoidal rule involves the second derivative of the function. The fourth derivative of \(x^3\) is 0. The error bound for the trapezoidal rule can be defined as \((b-a)^3/(12n^2) * M\), where M is the maximum value of the absolute value of the fourth derivative on the interval [a, b]. For our function and given interval [0, 2], the max value of the fourth derivative is 0. Therefore, no matter the values of 'a', 'b', 'n' and 'M', the error in the approximation using the Trapezoidal Rule will be 0.

Step 2: Applying the Simpson's Rule

Similar to the trapezoidal rule, we need the fourth derivative for Simpson's rule as well. The error in this method can be calculated by using formula \((b-a)^5/(180n^4) * M\). As calculated earlier, M, the maximum value of the fourth derivative on interval [a, b] is 0 which makes the error 0 by Simpson's rule as well.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a simple technique used in numerical integration to approximate the definite integral of a function. It is named because it approximates the region under the graph of the function as a series of trapezoids. The core idea is to break the interval into smaller sub-intervals of equal length, calculate the area of trapezoids over these sub-intervals, and sum up these areas. This method is practical when dealing with a function that is complicated to integrate manually.

This method works by estimating the function's value at the endpoints of each sub-interval and forming a trapezoid. The area of the trapezoid is then calculated using the formula:

• Area of a trapezoid = \( \frac{1}{2} \times \text{base} \times \text{sum of parallel sides} \)

For a function \( f(x) \) integrated from \( a \) to \( b \), we use the following trapezoidal rule formula:

• \( \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{2n} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \)

Here \( n \) is the number of sub-intervals, and \( x_0, x_1, ..., x_n \) are the points dividing the intervals. It's a straightforward implementation, but it’s important to check whether the error is acceptable. Error estimation for this rule involves the second derivative of the function, and in our specific exercise, it results in an error of zero due to the derivative being zero over the interval.

Simpson's Rule

Simpson's Rule is a more accurate approximation method used in numerical integration. It is based on the idea of fitting a parabola through pairs of points of the function, rather than just straight lines as in the Trapezoidal Rule. This allows Simpson's Rule to provide a better approximation over the same interval and with very few sub-intervals.

This technique requires an even number of sub-intervals, dividing the interval into sections typically with three points forming parabolic segments. The formula for Simpson's Rule can be expressed as:

• \( \int_{a}^{b} f(x) \, dx \approx \frac{b-a}{3n} \left[ f(x_0) + 4 \sum_{i=1,3,5,...}^{n-1} f(x_i) + 2 \sum_{i=2,4,6,...}^{n-2} f(x_i) + f(x_n) \right] \)

This rule employs both the odd and even terms, with odd terms multiplied by four and even terms by two. It captures more function features and smooths out curves in the data, providing a better fit.

Similar to the Trapezoidal Rule, the error estimation for Simpson's Rule in our exercise yielded zero, mainly because the higher derivative responsible for error estimation, the fourth derivative, was zero. This resulted in an exact approximation within the given bounds.

Error Estimation

Error estimation in numerical integration helps to evaluate the accuracy of the approximation techniques like the Trapezoidal and Simpson's Rules. It involves analyzing how much the estimated integral might deviate from the true value.

For the Trapezoidal Rule, the error can be estimated using the formula:

• \( E_T = -\frac{(b-a)^3}{12n^2}M \)

Here, \( M \) is the bound on the second derivative of the function over the interval \( [a, b] \). For Simpson's Rule, the error is described by:

• \( E_S = -\frac{(b-a)^5}{180n^4}M \)

In this context, \( M \) represents the maximum absolute value of the fourth derivative. These formulas indicate the error decreases with an increase in the number of sub-intervals \( n \).

In the original exercise, both integration methods revealed zero error due to the zero fourth derivative, ensuring the computations were quite precise. Understanding these error estimations is crucial because it allows checks on the validity and accuracy of numerical approximations, ensuring efficient application of these rules in real-world scenarios.