Estimating+Area+using+Rectangles+(left,+right,+midpoint)+and+Trapezoids

Integration is the method by which mathematitions find the area under the curve. Similarly, area can be found by using traditional geometric formulae, but in order to compute area, one must specify the parameters of the polygon.

Two such methods are common. One is by means of rectangles and is called the Riemann sum. The other is by utilizing trapezoids and is called a trapezoidal sum.

The heights of these geometric figures is obtained by taking the domain interval over which the area is being found and dividing this quantity by the number of rectangles desired with more rectangles resulting in a more accurate area. The same is true for trapezoids. Alternatively, the intervals over which the geometric figures are defined can be gained from a data table using defined points.

The Riemann sum is basic in that the length of the rectangle can either be the leftmost coordinate point, the rightmost coordinate point, or the midpoint of the two coordinate points. Multiplying length by the height of the rectangle will provide the area of a rectangle. Add the areas of all the rectangles in order to produce the estimated sum. The median Riemann sum is the most accurate as a left Riemann sum is too low when the function is increasing and too high when the function is decreasing, wheras the right Riemann sum is too high when the function is increasing and too low when the function is decreasing.

A trapezoidal sum is gained by multiplying the height by the average range of the left and right outputs. This average is the quantity of the sum of the range values divided by two.

A known shortcut for these methods is to factor out the height of the geometric figures, and occasionally the half involved in the trapezoidal function, but this method can only be utilized when all the heights are the same and the intervals are defined by a table of data of unequal domain intervals.