5 Steps to a 5 AP Calculus BC 2019

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