ABSTRACT
In this chapter, we will study various properties of the Riemann integral, some of which you have seen in elementary calculus courses. Throughtout, in what follows we shall assume that if a function f is defined on a “point interval” [c, c], then any Riemann sum consists of only one interval of length zero. Then, any Riemann sum is zero, and hence, the integral of f on the point interval should be zero, i.e., ∫ b a f ( x ) d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351180641/0424c007-6610-4c93-8ae3-088a74bd8b1d/content/eq1692.tif"/> . Similarly, if f is defined on a “backward interval” [b, a] with b > a, then the length of each subinterval is negative and thus the integral over the backward interval is the negative of the integral on its forward counterpart, i.e., ∫ b a f ( x ) d x = − ∫ a b f ( x ) d x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351180641/0424c007-6610-4c93-8ae3-088a74bd8b1d/content/eq1693.tif"/> .
