The Trapezoidal Rule is a numerical method used to approximate the definite integral of a function, by dividing the area under the curve into trapezoids and summing their areas. It is particularly useful for functions that are difficult to integrate analytically, providing a balance between simplicity and accuracy, especially when the function is relatively smooth over the interval.

Trapezoidal Rule

numerical method

The definite integral of a function over an interval is a fundamental concept in calculus that represents the accumulation of quantities, such as area under a curve, over that interval. It is evaluated using the limits of integration and the antiderivative of the function, often employing the Fundamental Theorem of Calculus to connect differentiation and integration.

definite integral

approximate the definite integral

The area under the curve (AUC) in a graph represents the integral of a function, providing a quantitative measure of the total accumulation of the quantity represented by the curve over a specified interval. It is widely used in various fields such as statistics, physics, and economics to analyze the cumulative effect or total value of a variable across a range of conditions.

area under the curve

trapezoids

summing their areas

functions that are difficult to integrate

Analytical integration

balance between simplicity and accuracy

A smooth function is a function that has derivatives of all orders, which means it is infinitely differentiable over its domain. This property ensures that the function behaves predictably and without abrupt changes, making it an essential concept in calculus and analysis for studying the behavior of curves and surfaces.

Smooth Function

An interval is a range of numbers between two set points, which can be either inclusive or exclusive of these endpoints. It is a fundamental concept in mathematics, particularly in calculus and statistics, used to describe the set of real numbers that lie between two boundaries.

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