Approximate Integral: Trapezoid & Simpson's Rule (n=6)

Let's dive into the world of numerical integration! Sometimes, finding the exact value of a definite integral can be tricky or even impossible using traditional calculus methods. That's where numerical methods like the Trapezoid Rule and Simpson's Rule come to the rescue. They provide excellent approximations, especially when dealing with complex functions. In this article, we'll explore how to approximate the definite integral $\int_{-2}^2 \frac{1}{\sin x+6} dx$ using both the Trapezoid Rule and Simpson's Rule with $n = 6$.

Understanding Numerical Integration

Before we jump into the calculations, let's briefly touch on the core idea behind numerical integration. Instead of finding the antiderivative and evaluating it at the limits of integration (which is the standard method for definite integrals), these methods approximate the area under the curve by dividing the interval into smaller subintervals and using geometric shapes to estimate the area within each subinterval. The Trapezoid Rule uses trapezoids, while Simpson's Rule employs parabolas for a more refined approximation.

The Trapezoid Rule: A Step-by-Step Guide

The Trapezoid Rule approximates the definite integral by dividing the area under the curve into trapezoids. It's like fitting a series of trapezoids under the curve and summing their areas. The formula for the Trapezoid Rule is:

$$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]$$

Where:

• $a$ and $b$ are the limits of integration.
• $n$ is the number of subintervals.
• $\Delta x = \frac{b - a}{n}$ is the width of each subinterval.
• $x_i = a + i\Delta x$ are the points dividing the interval.

Now, let's apply this to our integral $\int_{-2}^2 \frac{1}{\sin x+6} dx$ with $n = 6$.

1. Calculate $\Delta x$:

   $$\Delta x = \frac{2 - (-2)}{6} = \frac{4}{6} = \frac{2}{3}$$

2. Determine the $x_i$ values: We start at $x_0 = -2$ and increment by $\Delta x = \frac{2}{3}$:

   • $x_0 = -2$
   • $x_1 = -2 + \frac{2}{3} = -\frac{4}{3}$
   • $x_2 = -2 + 2(\frac{2}{3}) = -\frac{2}{3}$
   • $x_3 = -2 + 3(\frac{2}{3}) = 0$
   • $x_4 = -2 + 4(\frac{2}{3}) = \frac{2}{3}$
   • $x_5 = -2 + 5(\frac{2}{3}) = \frac{4}{3}$
   • $x_6 = -2 + 6(\frac{2}{3}) = 2$

3. Evaluate $f(x) = \frac{1}{\sin x + 6}$ at each $x_i$:

   • $f(x_0) = f(-2) = \frac{1}{\sin(-2) + 6} \approx 0.1709$
   • $f(x_1) = f(-\frac{4}{3}) = \frac{1}{\sin(-\frac{4}{3}) + 6} \approx 0.1764$
   • $f(x_2) = f(-\frac{2}{3}) = \frac{1}{\sin(-\frac{2}{3}) + 6} \approx 0.1826$
   • $f(x_3) = f(0) = \frac{1}{\sin(0) + 6} = \frac{1}{6} \approx 0.1667$
   • $f(x_4) = f(\frac{2}{3}) = \frac{1}{\sin(\frac{2}{3}) + 6} \approx 0.1512$
   • $f(x_5) = f(\frac{4}{3}) = \frac{1}{\sin(\frac{4}{3}) + 6} \approx 0.1583$
   • $f(x_6) = f(2) = \frac{1}{\sin(2) + 6} \approx 0.1463$

4. Apply the Trapezoid Rule formula:

   $$\int_{-2}^2 \frac{1}{\sin x+6} dx \approx \frac{\frac{2}{3}}{2} [0.1709 + 2(0.1764) + 2(0.1826) + 2(0.1667) + 2(0.1512) + 2(0.1583) + 0.1463]$$

   $$\approx \frac{1}{3} [0.1709 + 0.3528 + 0.3652 + 0.3334 + 0.3024 + 0.3166 + 0.1463]$$

   $$\approx \frac{1}{3} [1.9876] \approx 0.6625$$

Therefore, the approximate value of the definite integral using the Trapezoid Rule with $n = 6$ is approximately 0.6625.

Simpson's Rule: A More Accurate Approach

Simpson's Rule takes numerical integration to the next level by using parabolas to approximate the area under the curve. This generally provides a more accurate estimate compared to the Trapezoid Rule, especially for functions with curves. The formula for Simpson's Rule is:

$$\int_{a}^{b} f(x) dx \approx \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$$

Key things to note about Simpson's Rule:

• $n$ must be an even number.
• The coefficients alternate between 4 and 2, except for the first and last terms, which have a coefficient of 1.

Let's apply Simpson's Rule to our integral $\int_{-2}^2 \frac{1}{\sin x+6} dx$ with $n = 6$.

1. We already have $\Delta x$ and the $x_i$ values from the Trapezoid Rule calculation:

   • $\Delta x = \frac{2}{3}$
   • $x_0 = -2, x_1 = -\frac{4}{3}, x_2 = -\frac{2}{3}, x_3 = 0, x_4 = \frac{2}{3}, x_5 = \frac{4}{3}, x_6 = 2$

2. We also have the $f(x_i)$ values:

   • $f(x_0) \approx 0.1709$
   • $f(x_1) \approx 0.1764$
   • $f(x_2) \approx 0.1826$
   • $f(x_3) \approx 0.1667$
   • $f(x_4) \approx 0.1512$
   • $f(x_5) \approx 0.1583$
   • $f(x_6) \approx 0.1463$

3. Apply Simpson's Rule formula:

   $$\int_{-2}^2 \frac{1}{\sin x+6} dx \approx \frac{\frac{2}{3}}{3} [0.1709 + 4(0.1764) + 2(0.1826) + 4(0.1667) + 2(0.1512) + 4(0.1583) + 0.1463]$$

   $$\approx \frac{2}{9} [0.1709 + 0.7056 + 0.3652 + 0.6668 + 0.3024 + 0.6332 + 0.1463]$$

   $$\approx \frac{2}{9} [2.9904] \approx 0.6645$$

Therefore, the approximate value of the definite integral using Simpson's Rule with $n = 6$ is approximately 0.6645.

Comparing the Results

• Trapezoid Rule approximation: 0.6625
• Simpson's Rule approximation: 0.6645

As expected, Simpson's Rule provides a slightly more accurate approximation than the Trapezoid Rule for this integral. This is because Simpson's Rule uses parabolas to fit the curve, which generally provides a better fit than the trapezoids used in the Trapezoid Rule.

Conclusion

Numerical integration methods like the Trapezoid Rule and Simpson's Rule are powerful tools for approximating definite integrals, especially when analytical solutions are difficult or impossible to obtain. By dividing the area under the curve into smaller shapes and summing their areas, these methods provide accurate estimates that are widely used in various fields of science and engineering.

For further exploration and a deeper understanding of numerical integration techniques, you can visit Wikipedia's article on Numerical Integration. This resource provides comprehensive information, including various methods, error analysis, and applications.