TRAPEZOIDAL APPROXIMATION Meaning and
Definition
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Trapezoidal approximation is a numerical method used to estimate the value of a definite integral. It involves dividing the interval of integration into multiple subintervals and approximating the area under the curve using trapezoids. These trapezoids are formed by connecting the points on the curve with straight line segments, forming a series of trapezoidal shapes.
Each trapezoid represents an approximation of the small area under the curve within a given subinterval. The area of each trapezoid can be calculated by taking the average of the heights of the two vertical sides and multiplying it by the width of the subinterval. By summing the areas of all the trapezoids, an estimation of the total area under the curve can be obtained.
The accuracy of the trapezoidal approximation improves as the number of subintervals increases, leading to a better estimation of the definite integral. This method is based on the assumption that the function being integrated is approximately linear within each subinterval.
Trapezoidal approximation has various applications in fields like physics, engineering, and economics, where calculations involving definite integrals are necessary. It provides a simple yet effective numerical technique to approximate the value of a definite integral, particularly when an exact solution or evaluation is not feasible.
Etymology of TRAPEZOIDAL APPROXIMATION
The word "trapezoidal" is derived from the shape of the trapezoid, a geometrical figure with four sides, two of which are parallel. The term "approximation" refers to the process of estimating or calculating a value that is close to the actual value but not necessarily exact. Therefore, "trapezoidal approximation" refers to a method of approximating a value by using a trapezoid shape to estimate the area under a curve in calculus or to approximate a definite integral.
