Q:

# What is the midpoint method?

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The midpoint method is a means of approximating the integration of a function over a span of values. It is employed in calculus and is a more accurate way of determining the area under a curve than by simply using the left or right points.

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The midpoint method is based on the principle that if a function is calculating a rate of change (such as a velocity graph), then the area under a given segment of the curve of that function is the displacement based on that rate of change. For example, the rate of change of f(x)=x^2 is a constant line of 2x. The area under the line y=2x between x-values 0 and 2 is 4, which is the same as f(2).

However, not all rate-of-change functions, also called derivatives, are straight lines. Therefore, the area under the curve cannot be determined with a single operation. Area can be approximated using a series of rectangles taken at regular intervals and then added together. The width of the rectangles varies based on the number of subdivisions the individual wants to make; more rectangles over a given area mean more accuracy. The height of each rectangle is taken from the y-value of the derivative.

Instead of using the right or left point of each interval to find the height, the midpoint method uses the x-value in the middle of each interval. This helps account for any area lost or gained by giving close to an average of heights. A method that uses the average is, therefore, more accurate than the trapezoidal method.

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