Page 124 - Calculus Demystified

CHAPTER 4
b The Integral 111
that area is f(x) dx. We conclude therefore that
a
b

f(x) dx = F(b) − F(a).
a

y = f(x)

a b

_
F(b) F(a)

Fig. 4.10

Finally, if G is any other antiderivative for f then G(x) = F(x) + C. Hence

b

G(b) − G(a) =[F(b) + C]−[F(a) + C]= F(b) − F(a) = f(x) dx.
a
That is the content of the Fundamental Theorem of Calculus.

2
You Try It: Calculate the area below the curve y =−x + 2x + 4 and above the
x-axis.

4.3 SignedArea

Without saying so explicitly, we have implicitly assumed in our discussion of area in
the last section that our function f is positive, that is its graph lies about the x-axis.
But of course many functions do not share that property. We nevertheless would
like to be able to calculate areas determined by such functions, and to calculate the
corresponding integrals.
This turns out to be simple to do. Consider the function y = f(x) shown in
Fig. 4.11. It is negative on the interval [a, b] and positive on the interval [b, c].