Sec4-4: (Day1) Fundamental Theorem of Calculus
Sec4-4: #2-38 evens
Fundamental Theorem of Calculus: Makes a connection between
Indefinite Integrals (Antiderivatives) and Definite Integrals (“Area”)
Historically, indefinite integration has always been defined to be
the inverse of differentiation.
 f ( x)dx
is the collection of all possible anti-derivatives of
f(x), which happen to differ only by a constant.
But definite integration, motivated by the problem of finding areas
under curves, was originally defined as a limit of Riemann sums.

Is the limit of any Riemann sum as the number of
b
f ( x)dx rectangles approaches infinity … provided the
a
limit of the Lower and the limit of the Upper
Riemann sums are equal.
Only later was it discovered that the limits of these Riemann
sums can actually be computed with antiderivatives, leading to
our modern Fundamental Theorem of Calculus.
b
f ( x)dx  g (b)  g (a) if
_______________________________
a
and only if _________________
g ( x)  f ( x)
The fundamental theorem allows us to calculate definite integrals

b
a
f ( x)dx 
lim
(Riemann Sum)
n
By using anti-derivatives (indefinite integrals)

f ( x)dx  g ( x)

g ( x)  f ( x)
_________________
Examples: Applying the Fundamental Theorem of Calculus

b
a
f ( x)dx  g (b)  g (a) where g ( x)  f ( x)
Section 4-4
5.
11.
15.
25.

2
0

1
0

4

4
1
0
6 x dx
(2t  1) 2 dt
(u  2)
du
u
x 2  9 dx
Examples: Applying the Fundamental Theorem of Calculus

29.



33.
6




a
1sin2 ( )
cos 2 ( )
0

31.
4
b
f ( x)dx  g (b)  g (a) where g ( x)  f ( x)
d
sec 2 (x) dx
6
3
3
4sec(x)tan(x ) dx