Question

Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{2} x^{3} d x $$

Short answer

First, use the trapezoidal and Simpson's rules to find approximations for the integral. Then, use the formulas for error bounds to calculate the maximum error in these approximations. The maxima of the second and fourth derivatives of \( f \) are needed for these calculations.

Step-by-step solution

Trapezoidal rule approximation

Apply the trapezoidal rule with \( n = 4 \) for the integral. The formula for the trapezoidal rule is: \[ T_{n} = \frac{b-a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a+ i \frac{b-a}{n}) + f(b) \right] \] where \( f(x) = x^{3} \), \( a = 0 \), \( b = 2 \], and \( n = 4 \).

Trapezoidal rule error bound

Calculate the error bound for the trapezoidal rule. The formula for this error bound is: \[ |E_{T}| \leq \frac{(b-a)^{3}}{12n^{2}} M \] where \( M \) is the maximum value of the absolute value of the second derivative of \( f \) on the interval [a, b]. Calculate \( M \) first by taking the second derivative of \( f(x) \), then find its maximum on [a, b].

Simpson's rule approximation

Now apply the Simpson's rule with \( n = 4 \) for the integral. The Simpson's rule is: \[ S_{n} = \frac{b-a}{6n} \left[ f(a) + 4 \sum_{i=1,3,.., n-1}^{n-1} f(a+ i \frac{b-a}{n}) + 2 \sum_{j=2,4,.., n-2}^{n-2} f(a+ j \frac{b-a}{n}) + f(b) \right] \] Use same \( f(x) \), \( a \), \( b \), and \( n \) from step 1.

Simpson's rule error bound

Calculate the error bound for Simpson's rule. The formula for estimating this error is: \[ |E_{S}| \leq \frac{(b-a)^{5}}{180n^{4}} M_{s} \] where \( M_{s} \) is the maximum value of the absolute value of the fourth derivative of \( f \) on the interval [a, b]. Find \( M_{s} \) in the same manner as \( M \) in step 2.

Key concepts

Trapezoidal Rule

When dealing with numerical integration, the trapezoidal rule is a simple, yet powerful method to approximate the value of a definite integral. Picture this: you have the graph of a function, and you want to find the area under the curve between two points on the x-axis. The trapezoidal rule essentially breaks this area into a series of trapezoids, calculates their areas, and sums them up to give an approximation of the entire area under the curve.

The formula is expressed as:
\[ T_{n} = \frac{b-a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(a+ i \frac{b-a}{n}) + f(b) \right] \]
Here, \(n\) represents the number of trapezoids, \(a\) and \(b\) define the interval, and \(f(x)\) is the function you're integrating. The key is to choose an \(n\) that provides a good balance between accuracy and computational effort.

Simpson's Rule

For those looking for greater accuracy when approximating definite integrals, Simpson's rule is a step up from the trapezoidal rule. This method fits parabolas to sections of the curve and calculates the area under these parabolas to estimate the integral. Its fundamental difference lies in its ability to more closely follow the curve, especially when the function is smooth.

The general formula for Simpson's rule is as follows:
\[ S_{n} = \frac{b-a}{6n} \left[ f(a) + 4 \sum_{i=1,3,.., n-1}^{n-1} f(a+ i \frac{b-a}{n}) + 2 \sum_{j=2,4,.., n-2}^{n-2} f(a+ j \frac{b-a}{n}) + f(b) \right] \]
Here, the sum terms with coefficients of 4 and 2 correspond to the midpoints and endpoints of the subintervals determined by \(n\), respectively. Simpson's rule generally yields more accurate results, particularly for functions that are well-approximated by second-degree polynomials within each subinterval.

Error Formulas

After using numerical methods like the trapezoidal rule or Simpson's rule, it's crucial to understand the potential error in these approximations. Error formulas allow us to estimate the maximum error bound in our calculations, giving us a gauge of how reliable our numerical integration is.

Trapezoidal rule error is bounded by:
\[ |E_{T}| \leq \frac{(b-a)^{3}}{12n^{2}} M \]
for the trapezoidal rule, where \(M\) is the maximum value of the function's second derivative on the interval [a, b]. For Simpson's rule, the error is bounded by:
\[ |E_{S}| \leq \frac{(b-a)^{5}}{180n^{4}} M_{s} \]
where \(M_{s}\) represents the maximum of the fourth derivative on the interval. These formulas underline an important principle: the error decreases rapidly as you increase \(n\) (the number of subdivisions), but the rate of decrease is different for each rule. Understanding these can help you choose the most appropriate method for your needs.

Numerical Integration

Numerical integration is a cornerstone of mathematical computation, especially when exact integration is difficult or impossible. Numerical methods provide a powerful toolkit for approximating the integral of functions that are either too complex for analytical integration or when dealing with data points rather than functions. Both the trapezoidal and Simpson's rules are examples of numerical integration techniques, each with its strengths and ideal use cases.

The choice between which method to use can depend on factors such as the function's behavior, the desired level of accuracy, and computational resources. In practice, numerical integration allows scientists and engineers to solve real-world problems in fields ranging from physics to finance, where exact solutions are often unattainable, yet approximations can provide valuable insights and predictions.