We start by listing some properties that should seem obvious because the Riemann integral is understood as an area (mathematical).

Fact.

(i) Letfbe a Riemann integrable function on[ Ifa,b].on f≥ 0[ then .a,b],

Iffis continuous on[ anda,b]on f> 0[ then .a,b],(ii) Let

fbe a Riemann integrable function on[− Ifa,a].fis an odd function, then .

Iffis an even function, then .(iii) Let

fbe a Riemann integrable function on[ Ifa,b].m,Mare real numbers such thaton m≤f≤M[ thena,b],(iv) Let

fandgbe Riemann integrable functions on[ Ifa,b].on f≤g[ thena,b],(v) Let

fbe a Riemann integrable function on[ Then its absolute valuea,b].| is a Riemann integrable function onf|[ anda,b]

The two properties in (ii) follow from the symmetry of graph just by looking at a picture:

Also other kinds of symmetry can be useful at times, for instance the following result should be hardly surprising:

Indeed, since the graph of the sine function is symmetric,

the areas above and below the *x*-axis cancel each other out.

The comparison properties (iii) and (iv) should be clear from this picture:

While the above facts were basically just observations, the following properties are quite important:

Theorem.

(i) Letbe real numbers, let a<b<cfbe a function defined on the interval[ The functiona,c].fis Riemann integrable on [a,c] if and only if it is Riemann integrable on both[ anda,b][ then alsob,c];(ii) Let

fbe a Riemann integrable function on[ leta,b],kbe a real number. Then the functionkfis Riemann integrable on[ anda,b](iii) Let

fandgbe Riemann integrable functions on[ Then the functiona,b].is Riemann integrable on f+g[ anda,b]

The first property should be again clear from a picture:

The second property follows algebraically from the definition of the Riemann
integral. Indeed, the constant *k* can be factored out of the supremum
and infimum of *f* over individual segments, then out of the sums for
upper and lower limits and finally from the infima and suprema defining the
integral.

The third property is less obvious, since the supremum of a sum of two functions is definitely not equal to the sum of individual suprema. However, one has a suitable inequality there and it can be worked out. There is also a geometric argument, a curious reader may find the outline (along with some interesting tidbits concerning areas) here.