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The trapezoidal rule is a method used in numerical integration to estimate the value of definite integrals. It works by dividing the area under a curve into trapezoids, calculating the area of each, and then adding them up. This technique is especially useful when a function is difficult to integrate analytically.

To apply the trapezoidal rule, follow these steps:

• Divide the interval of integration into smaller, equal-width subintervals.
• Calculate the endpoints of each subinterval, known as nodes or points.
• Evaluate the function at each node.
• Use the formula:\[\int_a^b f(x) \, dx \approx \frac{h}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right)\]where \(h\) is the width of each subinterval.

In this exercise, with \(n=4\), the width \(h\) is calculated as \(\frac{\pi}{4}\) and function values are estimated at each node, then inserted into the formula to get an approximate value of the integral.

Simpson's rule is another technique for numerical integration that often provides a more accurate approximation than the trapezoidal rule, especially for functions that are relatively smooth. This rule combines the concepts of the trapezoidal rule with parabolic curves, attempting to better fit curvatures within the interval.

Here's how Simpson's rule works:

• The interval is divided into an even number of subintervals, just like with the trapezoidal rule.
• For each pair of subintervals, a quadratic polynomial is used, involving three points: the start, midpoint, and end.
• The function values at these points are used in the formula:\[\int_a^b f(x) \, dx \approx \frac{h}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_n) \right)\]where \(h\) is again the width of each subinterval.

For this exercise, using \(n=4\), function values again at the nodes are used to compute an estimate of the integral with this method, yielding a potentially more accurate result than the trapezoidal method.

A definite integral refers to the computation of the area under a curve over an interval \([a, b]\). It represents the accumulation of quantities, such as area, volume, or even physical properties like mass or energy, over that interval.

The notation for a definite integral is:\[\int_a^b f(x) \, dx\]where \(f(x)\) is the function being integrated, \(a\) is the lower limit, and \(b\) is the upper limit.

The value of a definite integral is a specific numerical value that represents the total accumulation over the interval from \(a\) to \(b\). It can often be difficult or impossible to find the exact value of a definite integral analytically, especially when the function involves complex expressions. That's where numerical methods, like the trapezoidal and Simpson's rules, become useful. These methods provide approximate solutions based on dividing the interval into smaller, more manageable pieces.

When dealing with numerical integration, the concept of subintervals is crucial. Subintervals divide the main interval of integration into smaller pieces, making it easier to estimate the area under the curve.

To create subintervals:

• Determine the total interval \([a, b]\) over which to integrate.
• Decide on the number of subintervals, \(n\), which impacts the approximation's accuracy.
• Compute the width of each subinterval, typically denoted as \(h\), using the formula \(h = \frac{b-a}{n}\).
• Identify the dividing points, or nodes, which are equally spaced if \(h\) is consistent. These points are critical for calculating function values in numerical methods.

In this problem, the interval \([0, \pi]\) is divided into four subintervals, resulting in \(h = \frac{\pi}{4}\). These subintervals enable efficient application of both the trapezoidal and Simpson's rules, allowing us to approximate the definite integral effectively.