Math 151
Section 6.4
The Fundamental Theorem of Calculus
Indefinite Integral The indefinite integral of f is used to indicate the process of finding the
antiderivative of f, i.e.,
! f ( x) dx = F ( x) + C .
Example: Evaluate the following integrals.
A.
! 2x
B.
! 3x
5
+ 7x + 4 dx
2
da
The Fundamental Theorem of Calculus, Part 1 If f is continuous on [a, b], then
b
" f ( x) dx = F (b) ! F (a) where F is any antiderivative of f.
a
Example: Evaluate the following integrals.
5
A.
! 3x
2
+ 4x + 2 dx
1
4
B.
! 3x + 8e
0
5
C.
! 3x
1
2
da
4x
dx
Math 151
!
4
D.
! sec( x) tan( x) dx
0
5
E.
"
!2
1
dx
x2
3
F.
"
x 2 ! 4 dx
0
The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then the function g
x
defined by g ( x) = ! f (t ) dt with a ! x ! b is continuous on [a, b], is differentiable on (a, b), and
a
g !( x) = f ( x) .
Example: Find g !( x) .
x
A. g ( x) = ! t 2 +1dt
a
x2
B. g ( x) = ! t 3 + 2 dt
4
Math 151
2
C. g ( x) = ! ln (u) du
x3
D. g ( x) =
x 3 +1
!
u5 + 2 du
x2
Example: Use the graph of f !( x) , shown below, to answer the following questions.
A. Which is greater, f (15) or f (25)?
B. If f (5) = 30, what is f (0)?
C. If f (5) = 30, what is f (20)?
Math 151
a
Example: Compute g(10) where g(a) is defined by g ( a) = ! 2 f ( x) dx and f (x) is the graph shown
0
below.
Example: A particle is moving in straight line motion and its velocity is described by the function
v (t ) = t 2 !t !6 (measured in meters per second).
A. Find the displacement from t = 1 to t = 4.
B. Find the total distance traveled from t = 1 to t = 4.