5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:1

Areas and Volumes 271

13.1 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 13.1-1.)

y f(t)

0 a x t

Figure 13.1-1

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 13.1-2.)

y

f(t)

a x t

0

Figure 13.1-2