David Loomis

is defined informally to be the signed area of the region in the xy-plane bounded by the graph of ƒ, the x- axis, and the vertical lines x = a and x = b. areagraph

Where F is the antiderivative of f

An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows thatboundedreal numbers

 If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus

 If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then

 If a > b then define

 If a is a real number thenreal number

 If c is any element of [a, b], thenelement