This concept of area function leads to the Fundamental Theorems of Integral
Calculus � First fundamental theorem of integral calculus
Let the area function be defined by A(x) = 
( )
x
∫a f x dx
 for all x ≥ a, where
the function f is assumed to be continuous on [a, b] Then A′(x) = f (x) for all
x ∈ [a, b] � Second fundamental theorem of integral calculus
Let f be a continuous function of x defined on the closed interval [a, b] and
let F be another function such that 
F( )
( )
d
x
f x
dx
=
 for all x in the domain of
f, then 
[
]
( )
F( )
C
F ( )
F ( )
b
b
a
a f x dx
x
b
a
=
+
=
−
∫