Calculating areas within irregular boundaries, such as along rivers or curved roads, is crucial in various fields, including surveying, engineering, and environmental management. Surveyors often begin by creating a traverse, a connected series of straight lines approximating the area's boundary. The coordinates of each traverse point are essential for calculating the enclosed area. The double meridian distance formula is a widely used technique for this purpose. This method utilizes the traverse's vertex coordinates to compute the total area within the approximate straight-line boundary, providing an efficient and reliable foundation for subsequent calculations. Additional tools and methods are used in the area between the traverse and the irregular boundary. 

A planimeter is a mechanical device designed to trace the irregular boundary on a scaled map. By integrating the enclosed area directly, the planimeter can achieve an accuracy of approximately one percent, making it a trusted tool for surveyors and engineers.

Numerical methods like the trapezoidal rule and Simpson's one-third rule are employed for enhanced flexibility and accuracy. The trapezoidal rule divides the boundary into segments and calculates the area of the trapezoids formed by offsets perpendicular to the baseline. By summing these areas, it approximates the total irregular region. 

In contrast, Simpson's one-third rule fits parabolic curves between offsets, applying a weighted formula to achieve higher accuracy. This method is particularly effective when the boundary curvature is significant.

Combining these approaches allows practitioners to efficiently and precisely compute areas for various irregular boundaries, ensuring accuracy and adaptability to complex real-world scenarios.