Problem 49 - Calculus Early Transcendentals

Question

Approximate the following integrals using Simpson's Rule. Experiment with values of \(n\) to ensure that the error is less than \(10^{-3}\). \(\int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x=\frac{2 \pi}{3}\)

Step-by-Step Solution

Answer

In this problem, we are given the integral \(\int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x\), and we approximate it using Simpson's Rule while ensuring that the error is less than \(10^{-3}\). Simpson's Rule involves approximating the definite integral of a function by the sum of the areas of parabolas formed by pairs of points along the curve. We find that when \(n=10\), the error of the approximation is less than \(10^{-3}\). Therefore, the approximation of the given integral using Simpson's Rule with \(n=10\) is as follows: \[ \int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x \approx \frac{\pi}{60} \left[ f(0) + 4\sum_{i=1}^{10} f((2i-1) \frac{\pi}{20}) + 2\sum_{i=1}^{9} f(2i \frac{\pi}{20}) + f(\pi) \right] \]

Step 1: 1. Explanation of Simpson's Rule

Simpson's Rule is a numerical integration technique that involves approximating the definite integral of a function by the sum of the areas of parabolas formed by pairs of points along the curve. The formula for Simpson's Rule is given by: \[ \int_{a}^{b} f(x) dx \approx \frac{b-a}{6n}\left[ f(a) + 4\sum_{i=1}^{n} f(a+(2i-1) h) + 2\sum_{i=1}^{n-1} f(a + 2ih) + f(b) \right] \] where \(a\) and \(b\) are the limits of integration, \(n\) is the number of subintervals (even number), and \(h\) is the step size given by \(h = \frac{b-a}{2n}\).

Step 2: 2. Given function

The given function to be integrated is \(\frac{4 \cos x}{5-4 \cos x}\). We will use this function to compute Simpson's Rule approximation.

Step 3: 3. Interval and step size definition

The given interval of integration is \([0, \pi]\). The step size, \(h\), is defined as \(h=\frac{b-a}{2n} = \frac{\pi}{2n}\). We will vary \(n\) and compute the error until we find a suitable value.

Step 4: 4. Approximation using Simpson's Rule

We will now approximate the integral using Simpson's Rule as follows: \[ \int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x \approx \frac{\pi}{6n} \left[ f(0) + 4\sum_{i=1}^{n} f((2i-1) \frac{\pi}{2n}) + 2\sum_{i=1}^{n-1} f(2i \frac{\pi}{2n}) + f(\pi) \right] \] We will now compute the approximation for increasing values of \(n\) until the error is less than \(10^{-3}\).

Step 5: 5. Determining adequate value of \(n\)

To calculate the error of the approximation, we will compare the approximation with \(\frac{2 \pi}{3}\) (the given value). We will increase \(n\) until the following condition is met: \[ \abs{ \frac{2 \pi}{3} - \text{approximation}} < 10^{-3} \] By performing the calculations and increasing \(n\), a computed value of \(n = 10\) satisfies the error condition. The approximation of the given integral using Simpson's Rule becomes: \[ \int_{0}^{\pi} \frac{4 \cos x}{5-4 \cos x} d x \approx \frac{\pi}{60} \left[ f(0) + 4\sum_{i=1}^{10} f((2i-1) \frac{\pi}{20}) + 2\sum_{i=1}^{9} f(2i \frac{\pi}{20}) + f(\pi) \right] \]

Key Concepts

Numerical Integration

Numerical integration is an essential concept in calculus used to approximate definite integrals, especially when it is difficult or impossible to find an exact analytical solution.
In many real-world scenarios, the function you need to integrate may not have a simple antiderivative. This is where numerical methods like Simpson's Rule come into play.

• Simpson's Rule is one such technique that creates a parabolic approximation of the area under a curve.
• It is particularly useful for calculating the integral of complex or non-continuous functions.
• The method involves dividing the integration interval into an even number of small subintervals.
• A parabola is then fitted over the values at the endpoints and midpoints of these subintervals.

Simpson's Rule balances accuracy and computational efficiency, making it preferable over more computationally intensive methods like numerical methods requiring a higher number of subintervals.

Error Estimation

Error estimation is a crucial aspect of numerical integration, providing insight into the accuracy of your approximation.
When using Simpson's Rule, the error can be minimized by adjusting the number of subintervals, denoted as \(n\).

• The error in Simpson's Rule approximation decreases as you increase \(n\), but only up to a practical limit where further increase doesn't yield significant accuracy improvement.
• In typical exercises, you calculate the approximation until the error is less than a certain threshold, such as \(10^{-3}\) in our current example.
• The error formula for Simpson's Rule involves higher-order derivatives of the function being integrated. However, complex calculations are usually avoided in practice by experimentation with \(n\).

By calculating the approximation multiple times with increasing \(n\), and comparing each result to the actual value, you can ensure that your estimate is sufficiently accurate.

Definite Integrals

Definite integrals represent the signed area under a curve between two specified points \(a\) and \(b\).
For the given exercise, the definite integral of the function \( \frac{4 \cos x}{5-4 \cos x} \) over the interval \([0, \pi]\) is sought.

• Definite integrals are fundamental in applications involving area, volume, central points, among many other physical and geometric concepts.
• They provide a numeric value that conveys more than just the function's behavior; they summarize the accumulation of quantities represented by the function over an interval.
• In many cases, the most efficient way to evaluate a definite integral is through numerical methods when the analytical approach is infeasible.

Understanding definite integrals via numerical techniques like Simpson's Rule allows not only for theoretical applications, but also practical ones where functions integrate irregular behavior or complex forms.