WOLFRAM|DEMONSTRATIONS PROJECT

The Fundamental Theorem of Calculus

x-value

f(x)

2

x

10

x-

2

x

6

x

If

f

is a continuous function on

[a,b]

and

F

is an antiderivative for

f

on

[a,b]

, then

b

∫

a

f(t)t=F(b)-F(a).

If we take

a=0

and

F(a)=0

for convenience, then

F(x)

is the area under the graph of

f

from

0

to

x

and

f(x)

is the derivative (slope) of

F(x)

. In the image above, the purple curve is

f(x)

—you have three choices—and the blue curve is

F(x)

.