3.1. PROPERTIES OF THE INTEGRAL 39

Theorem 3.1.1 Suppose F ′ (x) = f (x) where f is a continuous function on [a,b] . Then∫ b

af (x)dx = F (b)−F (a) (3.5)

Proof: Let ε > 0 be given and let P be a partition a = x0 < x1 < · · ·< xn = b such thatwhenever ŷi ∈ [xi−1,xi] , ∣∣∣∣∣

∫ b

af (x)dx−

n

∑i=1

f (ŷi)(xi− xi−1)

∣∣∣∣∣< ε (3.6)

Then from the mean value theorem, there exists yi ∈ (xi−1,xi) such that

F (b)−F (a) =n

∑i=1

(F (xi)−F (xi−1))

=n

∑i=1

F ′ (yi)(xi− xi−1) =n

∑i=1

f (yi)(xi− xi−1)

Let ŷi in 3.6 be equal to yi just described. Then with the above, this shows that∣∣∣∣∫ b

af (x)dx− (F (b)−F (a))

∣∣∣∣< ε

Since ε is arbitrary, this verifies 3.5. ■

Example 3.1.2 Find∫ 2

0 cos(t)dt.

Note that cos(t) = sin′ (t) and so the above integral is sin(2)− sin(0) = sin(2).

Example 3.1.3 Find∫ b

a αdx.

A function whose derivative is α is x→ αx. Therefore, this integral is αb−αa =α (b−a).

The integral∫ b

a f (t)dt has been defined when f is continuous and a< b. What if a> b?The following definition tells what this equals.

Definition 3.1.4 Let [a,b] be an interval and let f be piecewise continuous on [a,b] ormore generally Riemann integrable on this interval. Then∫ a

bf (t)dt ≡−

∫ b

af (t)dt

Observation 3.1.5 With the above definition,∫ b

a dx is linear satisfying∫ b

a(α f +βg)dx = α

∫ b

af dx+β

∫ b

agdx

if a < b or b < a. Also∫ b

a αdx = αb−αa if a < b or b < a.