258 STEP 4. Review the Knowledge You Need to Score High
12.1 The FunctionF(x)=
∫x
a f(t)dt
The Second Fundamental Theorem of Calculus defines
F(x)=
∫x
a
f(t)dt
and states that iff is continuous on [a,b], thenF′(x)=f(x) for every pointxin [a,b].
If f ≥0, thenF ≥0.F(x) can be interpreted geometrically as the area under the
curve off fromt=atot=x. (See Figure 12.1-1.)
y f(t)
0 a x t
Figure 12.1-1
If f <0,F <0,F(x) can be treated as the negative value of the area between the
curve off and thet-axis fromt=atot=x. (See Figure 12.1-2.)
y
f(t)
t
a x
0
Figure 12.1-2