The Left-Hand Rule (LHR) is a rudimentary numerical integration technique for approximating the area under the curve. This technique combines (integrate) the areas of rectangular regions. It is common, but not necessary, to first subdivide a given interval into equally spaced subintervals whose length is typically denoted as .

The height for each rectangular region is where is the left-hand endpoint on the interval, thus, the reason behind the mystery why this rule is known as the Left-Hand Rule.

where for .

Notice that and

The Right-Hand Rule (LHR) is a rudimentary numerical integration technique for approximating the area under the curve. This technique combines (integrate) the areas of rectangular regions. It is common, but not necessary, to first subdivide a given interval into equally spaced subintervals whose length is typically denoted as .

The height for each rectangular region is where is the right-hand endpoint on the interval, thereby, once again demystifying the mystery why this rule is known as the Right-Hand Rule.

where for .

Notice that and

The Midpoint Rule is a numerical integration technique that seeks to balance the error of over-approximating and under-approximating found within the Left-Hand and the Right-Hand Rule.

  where    is the midpoint of the interval

NOTE: The relationship between the Midpoint Rule, Trapezoidal Rule, and Simpson’s Rule is given by the following identity.

Error Term

where for

The Trapezoidal Rule is a numerical integration technique that approximates the area under the curve by integrating the areas of various trapezoids. The number of partitions may either be even or odd.

Error Term

where for

Also known as Simpson’s Rule is a numerical integration technique that improves upon the Trapezoidal Rule by utilizing the geometry of parabolic arcs. The number of partitions must be even.

Error Term

where for