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)
.