Section 5.3: The Fundamental Theorem of Calculus
Instructor: Ms. Hoa Nguyen
([email protected])
What is the Fundamental Theorem of Calculus?
The Fundamental Theorem of Calculus establishes a connection between the two branches
of calculus: diﬀerential calculus and integral calculus. Diﬀerential calculus (derivative) arose
from the tangent problem, where integral calculus arose from the area problem. Newton and
Leibniz exploited the relationship of derivative and integral in the Fundamental Theorem
of Calculus to compute areas and integrals without having to compute them as limits of
Riemann sums.
The ﬁrst part of the Fundamental Theorem of Calculus
Deﬁne an equation of the form
g
(
x
) =
Z
x
a
f
(
t
)
dt
where
f
is a continuous function on [
a, b
] and
x
varies between
a
and
b
. Here,
x
is a variable
upper limit in the integral; so,
g
(
x
) is a function of
x
. If
x
is a ﬁxed number, then the
integral
R
x
a
f
(
t
)
dt
is a deﬁnite number.
Consider any continous function
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 Fall '08
 Noohi
 Calculus, Derivative, Fundamental Theorem Of Calculus, Riemann

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