Problem 56 - Essential Calculus

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral. Let \(n=4\) and round your answers to four decimal places. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{8 x}{x^{2}+4} d x $$

Step-by-Step Solution

Answer

The results will be the numeric approximations of the definite integral from applying the Trapezoidal Rule and Simpson's Rule, calculated up to four decimal places and then verified graphically.

Step 1: Identify the Function and Interval

The Function given is \( f(x) = \frac{8x}{x^2 + 4} \) and the interval of integration is \([0, 4]\)

Step 2: Apply the Trapezoidal Rule

The Trapezoidal Rule approximation is given by \(T_n= \frac{b - a} {2n} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\). Here, `a` is the lower limit and `b` is the upper limit of the integral. `n` is the number of divisions of the interval chosen (here, \(n=4\)). So, first, calculate \( x_i = a + i*\frac{b-a}{n} \) for \( i = 0, 1, 2, ..., 4\). Then substitute \( x_i \) into \( f(x) \) to get \( f(x_i) \). Finally, substitute \( a, b, n, f(x_i) \) into the Trapezoidal Rule equation to evaluate the estimate of the integral.

Step 3: Apply Simpson's Rule

The Simpson's Rule approximation is given by \( S_n = \frac{b - a} {3n} [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)] \). As defined before, use the calculated \( x_i \) and substitute into the function, then into the Simpson's Rule formula.

Step 4: Round to four decimal places

Once both calculations for Trapezoidal Rule and Simpson's Rule are done, remember to round both of the results to four decimal places as asked in the problem.

Step 5: Verify with a Graphing Utility

Finally, use a graphing method to plot the original function and find the areas under the curve with the Trapezoidal and Simpson's Rule methods. The numerical results should match with the graphical analysis.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical technique to approximate the value of a definite integral. Instead of calculating an exact area under the curve, this method uses trapezoids to estimate the area. Here's how it works:

• Divide the interval \( [a, b] \) into \( n \) equal parts, where \( n \) is the number of divisions.
• Calculate the width of each division as \( h = \frac{b-a}{n} \).
• Identify the function values at each dividing point: \( f(x_0), f(x_1), ..., f(x_n) \).
• Use the formula: \(T_n = \frac{b-a}{2n} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)]\)

This formula averages the area of the first and last function values and doubles the interstitial values when multiplying by the width of each segment. This approach can provide an accurate approximation when \( n \) is large. Don't forget to round the final result to the specified decimal places, in this case, four.

Simpson's Rule

Simpson's Rule is another method for approximating the value of definite integrals. It's often more accurate than the Trapezoidal Rule for the same number of intervals.
To apply Simpson's Rule:

• Ensure that \( n \) is even, as this method requires pairing subintervals.
• Calculate each partition's width as \( h = \frac{b-a}{n} \).
• Evaluate the function at each subdivision, just like in the Trapezoidal Rule.
• Use the formula: \(S_n = \frac{b-a}{3n} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)]\)

This formula uses alternating coefficients of 4 and 2 between the first and last terms. This alternation captures more details of the curve, improving accuracy. As with the Trapezoidal Rule, rounding to four decimal places ensures precision.

Definite Integrals

Definite integrals represent the area under a curve within a given interval. When you calculate a definite integral from \( a \) to \( b \), you're essentially summing up infinite tiny areas from \( a \) to \( b \).
Let's break it down:

• The integral is denoted by \( \int_{a}^{b} f(x) \, dx \).
• The limits \( a \) and \( b \) signify the start and end of the interval over which you're integrating.
• The function \( f(x) \) is the curve whose area you're finding under the \( x \)-axis.

When it's tough to find the exact integral using calculus due to complex functions, approximation methods like the Trapezoidal and Simpson's Rules become handy. These numerical methods provide a practical way to gauge the integral's value without needing the exact antiderivative.