176 CHAPTER 7. THE INTEGRAL
If f ≥ 0,a < b, then the mean value theorem implies that for F ′ = f , and some
t ∈ (a,b) ,F (b)−F (a) =∫ b
af dx = f (t)(b−a)≥ 0.
Thus ∫ b
a(| f |− f )dx ≥ 0,
∫ b
a(| f |+ f )dx ≥ 0 so∫ b
a| f |dx ≥
∫ b
af dx and
∫ b
a| f |dx ≥−
∫ b
af dx
so this proves∣∣∣∫ b
a f dx∣∣∣ ≤ ∫ b
a | f |dx. This, along with part 2 implies the other claim that∣∣∣∫ ba f dx
∣∣∣≤ ∣∣∣∫ ba | f |dx
∣∣∣ even if a > b.
The last claim is obvious because an antiderivative of 1 is F (x) = x.The change of variables theorem is available from the chain rule because if F ′ = f , then
f (g(x))g′ (x) = ddx F (g(x)) so that, from the above proposition,
F (g(b))−F (g(a)) =∫ g(b)
g(a)f (y)dy =
∫ b
af (g(x))g′ (x)dx.
We also have the integration by parts formula from the product rule. Say F ′ = f ,G′ = g.Then from the product rule, (FG)′ = f G+gF. In particular, if f ,g are continuous on [a,b] ,
F (b)G(b)−F (a)G(a) =∫ b
af (t)G(t)dt +
∫ b
ag(t)F (t)dt
These formulas are discussed more later.
Definition 7.1.5 A function f : [a,b] → R is piecewise continuous if there is anordered list of intermediate points zi having an order consistent with [a,b] , meaning thatzi−1 − zi has the same sign as a − b, a = z0,z1, · · · ,zn = b, called a partition of [a,b] ,and functions fi continuous on [zi−1,zi] such that f = fi on (zi−1,zi). For f piecewisecontinuous, define ∫ b
af (t)dt ≡
n
∑i=1
∫ zi
zi−1
fi (s)ds
If such a function fi exists, then it is uniquely defined on [zi−1,zi] as fi (zi)≡ limx→zi− f (x)with a similar definition for fi (zi−1).
Observation 7.1.6 Note that this actually defines the integral even if the function hasfinitely many discontinuities and that changing the value of the function at finitely manypoints does not affect the integral.
Of course this gives what appears to be a new definition because if f is continuous on[a,b] , then it is piecewise continuous for any such partition. However, it gives the sameanswer because, from this new definition,
∫ ba f (t)dt = ∑
ni=1 (F (zi)−F (zi−1)) = F (b)−
F (a).