Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=2 $$

Step-by-Step Solution

Answer

For approximation of the integral: \(\int_{0}^{2} e^{-x^2} dx \), (a) The Trapezoidal Rule gives approximately \( \(\frac{1}{2}\)(1 + 2e^{-1} + e^{-4}) \). (b) Simpson's Rule gives approximately \( \(\frac{1}{3}\)(1 + 4e^{-1} + e^{-4})\). Please round the answer to three significant digits.

Step 1: Divide the interval

For both the Trapezoidal Rule and Simpson’s Rule, firstly divide the interval from \(0\) to \(2\) into \(n=2\) subintervals, each of length \((2-0)/2 = 1\). So, the divisions would be \( x_0 = 0, x_1 = 1, x_2 = 2\).

Step 2: Evaluate the function at these points

Evaluate the function at these points, i.e. \( f(x_0) = e^{-x_0^2} = e^0 = 1 \), \( f(x_1) = e^{-x_1^2} = e^{-1} \), and \( f(x_2) = e^{-x_2^2} = e^{-4} \).

Step 3: Apply Trapezoidal Rule

The Trapezoidal Rule is defined as \( T_n = \(\frac{b - a}{2n}\)[f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)\]. Applying this formula, we obtain \( T_2 = \(\frac{2 - 0}{2*2}\)[f(x_0) + 2f(x_1) + f(x_2)] = \(\frac{1}{2}\)(1 + 2e^{-1} + e^{-4}) \).

Step 4: Apply Simpson's Rule

Simpson's Rule is defined as \( S_n = \(\frac{b - a}{3n}\)[f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)\]. As \( n = 2 \) is even, thus we can apply Simpson's rule, which gives us \( S_2 = \(\frac{2 - 0}{3*2}\)[1 + 4e^{-1} + e^{-4}] = \(\frac{1}{3}\)(1 + 4e^{-1} + e^{-4})\).

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a method used for numerical integration, allowing us to approximate the integral of a function over a specific interval. This method simplifies the area under a curve into a series of trapezoids and summates their areas.
The main idea is to split the interval \(a, b\) into smaller segments (determined by the number \(n\)), calculate the function's value at these points, and then approximate the area using trapezoids.
For our exercise, we split the interval \(0\, \text{to}\, 2\) into \(n = 2\) segments, each of width \(1\).
Here are the steps simplified:

• Evaluate the function: \(f(x_0) = e^{-0^2} = 1\), \(f(x_1) = e^{-1}\), and \(f(x_2) = e^{-4}\).
• Apply the Trapezoidal formula:

The rule's formula is: \[T_n = \frac{b - a}{2n}\left[f(x_0) + 2f(x_1) + \ldots + 2f(x_{n-1}) + f(x_n)\right]\]Replacing the values, we calculate \(T_2 = \frac{1}{2}\left[1 + 2e^{-1} + e^{-4}\right]\). This allows a simple computation, even though perfection is not guaranteed. The Trapezoidal Rule is handy for estimating integrals when exact solutions are complex or impossible to derive.

Simpson's Rule

Simpson's Rule is another numerical integration technique that provides more precise approximations than the Trapezoidal Rule by fitting a parabola through the data points rather than trapezoidal sections.
Ideal for smooth, continuous functions, this method divides the integration interval into an even number of subintervals and then applies a weighted average of the function's values. Our example divides \(0\, \text{to}\, 2\) into \(2\) segments:

• Evaluate the function at these points: \(f(x_0) = e^{-0^2} = 1\), \(f(x_1) = e^{-1}\), and \(f(x_2) = e^{-4}\).
• Use the Simpson's formula:

The formula is given by:\[S_n = \frac{b - a}{3n}\left[f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 4f(x_{n-1}) + f(x_n)\right]\]By using this formula with \(n=2\), we replace and compute \(S_2 = \frac{1}{3}\left[1 + 4e^{-1} + e^{-4}\right]\).
This approach yields more precision by utilizing a combination of the function evaluations, effectively improving on the basic estimation of the polynomial curve's area.

Approximate Integration

Approximate integration entails calculating the integral of a function when finding an exact solution may be difficult or impossible with standard analytical methods.
It provides flexibility by using various numerical techniques to estimate these integrals within a desired level of accuracy.
Techniques like the Trapezoidal Rule and Simpson's Rule offer practical solutions for calculating areas under curves, especially with complex functions. Each technique has its applications:

• The Trapezoidal Rule converts the area under a curve into trapezoids and is relatively straightforward but can lack accuracy.
• Simpson's Rule utilizes parabolic segments for a better approximation, especially with smoother curves, offering enhanced precision.

Appropriate technique selection depends on the problem at hand, balancing computational power and required accuracy. Using approximate integration, these methods empower calculations across diverse scientific and engineering domains, providing credible results even when precision isn't paramount.