Calculate the indicated areas. All data are accurate to at...

Calculate the indicated areas. All data are accurate to at least two significant digits. The widths of a cross section of an airplane wing are measured at $1.00-\mathrm{ft}$ intervals, as shown in Fig. $2.108 .$ Calculate the area of the cross section, using Simpson's rule. (FIGURE CAN'T COPY)

Calculate the indicated areas. All data are accurate to at least two significant digits.
The widths of a cross section of an airplane wing are measured at $1.00-\mathrm{ft}$ intervals, as shown in Fig. $2.108 .$ Calculate the area of the cross section, using Simpson's rule. (FIGURE CAN'T COPY)

Key Concepts

Significant Figures

Significant figures represent the precision of the measured or computed data. When numerical methods are applied, maintaining the correct number of significant digits is important for ensuring that approximations and final results reflect the accuracy of the original data.

Evenly Spaced Data

A key prerequisite for applying Simpson's rule effectively is that the data or sample points must be equally spaced across the interval. This uniform spacing ensures that the method's weighted average approach accurately captures the contribution of each subinterval.

Composite Simpson's Rule

When the integration interval is divided into multiple subintervals, Composite Simpson's Rule is used to approximate the entire integral. This method applies Simpson’s rule to each pair of subintervals and sums the results, allowing for more accurate approximations over a broader interval.

Simpson's Rule

Simpson's rule is a numerical integration technique that uses quadratic polynomials to approximate the integrand over subintervals. It requires an even number of subintervals, and its application typically results in an approximation that is more accurate than that of some other methods like the trapezoidal rule, particularly when the function being integrated is smooth.

Numerical Integration

Numerical integration involves techniques for approximating the value of definite integrals, particularly when an antiderivative is difficult or impossible to obtain in closed form. It is crucial for solving practical problems that involve the accumulation of quantities, especially when only discrete data points are available.

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