The Riemann sum of f over I with partition P is defined as
where xi−1 ≤ x*i ≤ xi. The choice of x*i in this interval is arbitrary. If x*i = xi−1 for all i, then S is called a left Riemann sum. If x*i = xi, then S is called a right Riemann sum. If x*i = 1⁄2(xi+xi−1), then S is called a middle Riemann sum. The average of the left and right Riemann sum is the trapezoidal sum.
If it is given that
where vi is the supremum of f over, then S is defined to be an upper Riemann sum. Similarly, if vi is the infimum of f over, then S is a lower Riemann sum.
Any Riemann sum on a given partition (that is, for any choice of x*i between xi−1 and xi) is contained between the lower and the upper Riemann sums. A function is defined to be Riemann integrable if the lower and upper Riemann sums get ever closer as the partition gets finer and finer. This fact can also be used for numerical integration.
Read more about Riemann Sum: Methods, Example, Animations
Famous quotes containing the word sum:
“To die proudly when it is no longer possible to live proudly. Death freely chosen, death at the right time, brightly and cheerfully accomplished amid children and witnesses: then a real farewell is still possible, as the one who is taking leave is still there; also a real estimate of what one has wished, drawing the sum of ones lifeall in opposition to the wretched and revolting comedy that Christianity has made of the hour of death.”
—Friedrich Nietzsche (18441900)