15.4 Definite Integrals and the Fundamental Theorem of Calculus
If
)
(
x
f
is continuous on [
a
,
b
], then:
)
(
)
(
)
(
a
F
b
F
dx
x
f
b
a
−
=
∫
where
F
is the general antiderivative of
f
(note that there is no arbitrary constant
C
added to definite integrals).
This formula is known as the (1
st
) Fundamental Theorem of Calculus.
From a graphical standpoint, this integral
represents the area of the region below
)
(
x
f
but above the
x
-axis if
0
)
(
≥
x
f
for all
x
on [
a
,
b
] or -1 times the
area of the region above
)
(
x
f
but below the
x
-axis if
0
)
(
≤
x
f
for all
x
on [
a
,
b
].
dx
x
f
b
a
)
(
A
∫
=
dx
x
f
b
a
)
(
A
∫
−
=

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