What Is the Midpoint Rule in Calculus?

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The midpoint rule of calculus is a method for approximating the value of the area under the graph during numerical integration. This is one of several rules used for approximation during numerical integration.

The midpoint rule of calculus has several general steps. Consider a formula that takes the value y = b(x) over an interval of x1 to x2, where x2 is equal to x1 + h. The value h refers to the distance on the x-axis between the start and end points. The midpoint rule with these values is equal to hf (x1 + h/2).

There is a variation to the midpoint rule that is also used in calculus, referred to as the composite midpoint rule. This variation approximates the value of the area under the graph by summing a series of rectangles. The intervals of this variation are found by dividing the distance between x1 and x2 into equal amounts. This formula is equal to h = (x2 – x1)/m, where m is the number of subintervals.

There are several other methods of numerical integration that are used to approximate the area under the graph. The triangle and trapezoid rules both use the summation of their respective shapes under the graph to approximate the area.