Problem 10 - Calculus: Early Transcendental Functions

Question

Use a computer or calculator to compute the Midpoint, Trapezoidal and Simpson's Rule approximations with \(n=10, n=20\) and \(n=50 .\) Compare these values to the approximation given by your calculator or computer. $$\int_{0}^{3} e^{-x^{2}} d x$$

Step-by-Step Solution

Answer

The exact answers require a numerical analysis software or a calculator capable of numerical integration, and might slightly differ depending on the tool used. Thus a final answer can't be given here. However the approximation values from the three methods should get closer to the true integral value as \(n\) increases.

Step 1: Computing the Midpoint Rule Approximation

For the Midpoint Rule, we distribute the interval \([0,3]\) into \(n\) equal segments, find the midpoint of each and evaluate the function at the midpoint. We sum up the results and multiply by the width of the intervals to achieve an approximation for the integral. This should be repeated for \(n=10, n=20\) and \(n=50\).

Step 2: Computing the Trapezoidal Rule Approximation

For the Trapezoidal Rule, we also divide the interval \([0,3]\) into \(n\) equal segments, and find the left and right endpoints for each interval. We then calculate the average of the function outputs for these two endpoints, then divide this by \(n\). This is likewise done for \(n=10, n=20\) and \(n=50\).

Step 3: Computing the Simpson's Rule Approximation

Simpson's Rule requires the interval \([0,3]\) to be divided into \(n\) equal segments where \(n\) is even. Within each segment, the function output at the left endpoint, four times the function output at the midpoint, and the function value at the right endpoint are calculated. These are multiplied by the width of the intervals and divided by 6 to give an approximation for each segment. The segment approximations are summed up to provide the overall Simpson's Rule approximation for the integral. This process is repeated for \(n=10, n=20\) and \(n=50\).

Key Concepts

Midpoint Rule

The Midpoint Rule is a numerical technique used to estimate the integral of a function over a specific interval. This method revolves around dividing the interval into equal parts. To understand it fully, consider these key points:

• Divide the interval \([a, b]\) into \(n\) equal segments.
• Determine the midpoints of each segment.
• Evaluate the function at these midpoints.
• Multiply the sum of these evaluations by the width of the segments.

In our context, we are working with the interval \([0, 3]\) and need to apply this rule with different segment numbers like 10, 20, and 50. The formula used for this approximation is:
\[M_n = \Delta x \sum_{i=1}^{n} f\left(m_i\right),\]
where \(\Delta x\) is the width of each segment and \(m_i\) represents the midpoint of the \(i\)-th segment. By leveraging this rule, you lean on the intuitive notion that the midpoint can fairly represent the average height of a function over an interval.

Trapezoidal Rule

The Trapezoidal Rule is a straightforward numerical integration technique that approximates the area under a curve as a series of trapezoids. Here's how you can use it:

• First, divide the interval \([a, b]\) into \(n\) equal segments.
• Identify the left and right endpoints of each segment.
• Compute the average of the function's values at these endpoints.
• Multiply this by the width of the segments to approximate the area under the curve.

For the function \(e^{-x^2}\) on the interval \([0, 3]\), this rule is based on the familiar formula:
\[T_n = \frac{\Delta x}{2} \left(f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n)\right),\]
where \(\Delta x\) is the width of each segment, and \(x_0, x_1, \, \ldots\,, x_n\) are the endpoints. The Trapezoidal Rule is advantageous due to its simplicity and ability to provide a good approximation when \(n\) is sufficiently large.

Simpson's Rule

Simpson's Rule provides a highly accurate estimation of the area under a curve by using parabolic segments instead of linear ones. This means it fits quadratic curves to the segments of the function, leading to improved precision.

To apply Simpson's Rule, follow these steps:

• Ensure \(n\) is even, with segments dividing the interval \([a, b]\) into equal parts.
• For each segment, calculate the function value at the left endpoint, four times the function value at the midpoint, and the value at the right endpoint.
• Multiply the sum by the width of the segment, then divide by 6.

This process is summed over all segments for the complete integral approximation. The formula is:
\[S_n = \frac{\Delta x}{3} \left(f(x_0) + 4 \sum_{\text{odd } i=1}^{n-1} f(x_i) + 2 \sum_{\text{even } j=2}^{n-2} f(x_j) + f(x_n)\right).\]
For the interval \([0, 3]\), Simpson's Rule generally offers greater accuracy than the Midpoint and Trapezoidal Rules, especially when dealing with smooth functions, due to its use of quadratic approximations.