Evaluate the integral of the following tabular data with (a)...

Evaluate the integral of the following tabular data with (a) the trapezoidal rule and (b) Simpson's rules:
$$\begin{array}{c|cccccc}
x & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 \\
\hline f(x) & 1 & 8 & 4 & 3.5 & 5 & 1
\end{array}$$

Key Concepts

Numerical Integration

This is the process of approximating the value of a definite integral when an analytical solution is challenging or impossible to obtain. It involves estimating the area under a curve by using discrete sets of data points or approximating functions, and is especially useful in practical situations where functions are given in tabular form or are complex.

Trapezoidal Rule

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing the interval into smaller subintervals and replacing the curve over each subinterval with a straight line. The area under each segment is calculated as the area of a trapezoid, which is then summed over all intervals. It is simple to implement and useful for uniformly spaced data, though it may be less accurate for functions that are highly nonlinear between data points.

Simpson's Rule

Simpson's rule is a more accurate numerical integration method that approximates the function on each subinterval by fitting a quadratic polynomial (a parabola) through three consecutive points. This technique generally provides a better approximation than the trapezoidal rule for smooth functions and requires that the number of subintervals be even. It is especially effective when the underlying function is well-approximated by second-degree polynomials over the intervals.

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