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Numerical Integration is a powerful technique used to approximate the integral of a function, especially when finding the exact integral is difficult or impossible. There are different methods, and one of them is the Trapezoidal Rule. This method divides the area under a curve into smaller trapezoids rather than using rectangles, which can often provide better approximations.
The basic idea of numerical integration is to replace a complex integral with a sum of simpler calculations. In our exercise, instead of finding the exact area under the curve of the function \(y=x \sqrt{\frac{4-x}{4+x}}\) from \(x=0\) to \(x=4\), we used the Trapezoidal Rule with 10 subintervals.
When implementing numerical integration techniques like this one, it's crucial to ensure that the method and number of subintervals chosen are suitable for capturing the behavior of the function being integrated.

The concept of Approximate Area underlies methods such as the Trapezoidal Rule used in numerical integration. Instead of calculating the exact area under a curve—which can sometimes be analytically impossible or require complex calculations—approximations are made.
In the Trapezoidal Rule, the curve is divided into small sections, each of which is roughly estimated using trapezoids. The sum of these trapezoidal areas provides an estimate for the total area under the curve. In our example, the approximate area was determined by calculating the area of each trapezoid formed between the integration boundaries \(x=0\) and \(x=4\).
By using smaller subintervals—or more trapezoids—we can often improve this approximation, getting closer to the true area. That's why choosing the number of subintervals (or \(n\)) is very important for achieving an accurate approximation.

Determining the Subinterval Width is a critical step in applying the Trapezoidal Rule for numerical integration. This width, often denoted by \(h\), is simply the division of the range of integration into smaller, equal parts. These parts form the base of the trapezoids used in the approximation.
To calculate \(h\), we use the formula:

• \(h = \frac{b - a}{n}\)

where \(a\) and \(b\) are the integration boundaries, and \(n\) is the number of trapezoids or subintervals. In the context of our exercise, the integration boundaries \(x=0\) and \(x=4\) are divided into 10 subintervals, resulting in a width \(h = 0.4\).
Choosing the appropriate \(h\) is vital. A smaller \(h\), which corresponds to more subintervals, usually provides a better approximation of the area but also requires more calculations.

Setting Integration Boundaries is the first step in numerical integration processes like the Trapezoidal Rule. These boundaries \([a, b]\) define the segment of the function over which the integral is approximated.
In our problem, the integration boundaries are given as \(x=0\) and \(x=4\), indicating that the area under the function \(y=x \sqrt{\frac{4-x}{4+x}}\) needs to be computed from \(x=0\) to \(x=4\). These boundaries effectively determine where the integration will start and stop, and they serve as limits in both the integral and in setting up the subintervals for numerical approximation methods.
Understanding and correctly identifying these boundaries is key to solving integration problems accurately. Incorrect boundaries can lead to wrong area approximations and affect the results significantly.