AP AB Calculus
Lesson 60: Fundamental Theorem of Calculus
Students will be able to…
 State and apply the fundamental theorem of calculus
 Use antiderivatives to evaluate definite integrals
Section 6.6 The Fundamental Theorem of Calculus
Theorem: The Fundamental Theorem of Calculus, Part I
b
 f ( x)dx  F (b)  F (a)
If f is continuous on [a,b] and if F is any antiderivative of f on [a,b], then
a
Proof:
Let x1, x2, ……., xn-1 be any points in [a,b] such that a < x1 < x2 < ….< xn-1 < b
These points divide [a,b] into n subintervals [a, x1], [x1,x2], …….[xn-1, b] whose lengths, as usual we denote by
x1, x2, ….xn
By hypothesis, F ( x)  f ( x) for all x in [a,b], so F satisfies the hypothesis of the Mean Value Theorem on each subinterval
mentioned above. Hence by the Mean-Value Theorem, we can find points
*
*
*
x1 , x 2 ,....x n in the respective subintervals such that
F ( x1 )  F (a )  F ( x1 )( x1  a )  f ( x1 )x
*
*
F ( x 2 )  F ( x1 )  F ( x 2 )( x 2  x1 )  f ( x 2 )x 2
*
*
F ( x 3 )  F ( x 2 )  F ( x 3 )( x 3  x 2 )  f ( x 3 )x 3
*
*
F (b)  F ( x n 1 )  F ( x n )( x n  x n 1 )  f ( x n )x n
*
*
Adding the preceding equations yields
n
F (b)  F (a)   f ( x k )x k
*
k 1
Let us now increase n in such a way that max x k  0. Since f is assumed to be continuous, the right side of the above equation
b
approaches
 f ( x)dx . But the left side is a constant independent of n; thus
a
F (b)  F (a) 
b
n
lim  f ( x
max x 0 k 1
k
)x k   f ( x)dx
a
b
F (b)  F (a) denoted by F(x)] a can be written as
 f(x)dx
 F(x)] a
a
b
Also can be written as  f(x)dx  F ( x)a and
a
b
 f(x)dx
 F ( x)] b x  a .
a
In words, the definite integral can be evaluated by finding any antiderivative of the integrand and then subtracting the value of this
antiderivative at the lower limit of integration from its value at the upper limit of integration.
Examples
2
1.
 x dx
1
2.
Find the area under the curve y = cos x over the interval [0, /2]
9
3.

x dx
1
ln 3
4.
 5e
x
dx
0
2
5.
1
 x dx
1
1
6.
1
 x dx
2
 /4
7.
 sec x tan x dx
 / 4
2
8.
 x dx
1
6
9.

0
x 2 , x  2
f ( x)dx if f(x)  
3x  2, x  2
1
10.
x
2
dx
1
0
11.
 x dx
4
b
There is a close relationship between the integrals
 f ( x) dx
and
 f(x) dx
a
However, the definite integral and the indefinite integral differ in some important ways.
The definite integral is a number ( the signed area between the graph of y = f(x) and the interval [a,b].
The indefinite integral is a function, or more accurately a set of functions [ the antiderivatives of f(x) ].
In an indefinite integral, the variable of integration is “passed through” to the antiderivatives in the sense that integrating a function of
x produces a function of x, integrating a function of t produces a function of t and so forth.
In a definite integral, the variable of integration is not passed through to the end result, since the end results in a number. Because the
variable of integration in a definite integral plays no role in the end result, it is often referred to as a dummy variable.
Theorem: Mean Value Theorem for Integrals
If f is continuous on a closed interval [a,b], then there is at least one number x * in [a,b] such that
b
 f ( x) dx  f ( x
*
)(b  a)
a
Example
12. Given f(x) = x2 over [1,4] find the value of x* guaranteed by the mean value theorem for integrals.
Theorem: The Fundamental Theorem of Calculus, Part 2
If f is continuous on an interval I, then f has an antiderivative on I. In particular, if a is any point in I, then the
x
function F defined by
F ( x)   f (t )dt is an antiderivative of f on I; that is, F ( x)  f ( x) for each x in I, or in an alternative
a

d 
  f (t )dt   f ( x)
dx  a

x
notation
If a definite integral has a variable upper limit of integration and a continuous integrand then the derivative of the integral with respect
to its upper limit is equal to the integral evaluated at the upper limit.
Examples
13.
x
d  3 
 t dt 
dx  1

14.
x

d 
  sin t dt 
dx  1

x
 (t
15. Given F(x) =
3
 1)dt , find F(1) and F (x)
1
x
16. Let
F ( x)   t 2  1 dt find F(3) and F (3)
F (3)
3
The two parts of the Fundamental Theorem of Calculus, when taken together, tell us that differentiation and integration are inverse
processes in the sense that each undoes the effect of the other.
It is common to treat parts 1 and 2 of the Fundamental Theorem of Calculus as a single theorem, and refer to it
simply as the Fundamental Theorem of Calculus. This theorem ranks as one of the greatest discoveries in the history of science, and
its formulation by Newton and Leibniz is generally regarded to be the “discovery of calculus.”
d
dx
NOTE
17. find
d
dx
g ( x)
 f (t ) dt  f ( g ( x))  g ( x)
a
sin x
t
2
dt
3
g ( x)
d
f (t ) dt  f ( g ( x))  g ( x)  f (h( x)  h ( x)
dx h (x )
sin x
18. find
d
t 2 dt

dx 3 cos x
Sample AP Problems
x
1.
Let
f ( x)   t 3 dt. Find the value of f ( 2).
0
10
2.
Let
f ( x)   t 3 dt. Find the value of f ( 2).
x
3x
3.
Let F ( x ) 
t
 e dt. Find the value of F (0).
4
2
4.
Let f be a function whose domain is the closed interval [0,5]. The graph of f is shown below
x
3
2
Let h(x) =
 f (t )dt
0
a) find the domain of h.
b) Find h ( 2).
c) At what x is h(x) a minimum? Show the analysis that leads to your conclusion.
5.
The following graph of f consists of line segments and semicircles. Use it to evaluate the following
integrals
14
a)
 f ( x)dx
0
10
b)
 f ( x)dx
0
12
c)
 f ( x)dx
3
4
6.
Given functions f and g, use

0
evaluate the following integrals
4
a)
 (3  f ( x)  1)dx
0
6
b)
 f ( x)dx
0
6
c)
 [ f ( x)  4  g ( x)dx
0
2
d)
 g ( x)dx
0
6
f ( x)dx  4,

4
6
f ( x)dx  3, and
 g ( x)dx  5 , and
0
6
 g ( x)dx  4 to
2
7. The graph of a function f consists of a semicircle and two line segements as shown below. Let g be the function given by
x
g ( x)   f (t )dt.
0
a)
b)
c)
d)
Find g(3).
Find all values of x on the open interval (-2,5) at which g has a relatiave maximum. Justify your answer.
Write an equation for the tangent to the graph of g at x = 3.
Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2,5). Justify your answer.
x
8. Let g(x) =
 h(t )dt ,
where h is the function whose graph is shown below
2
a) Evaluate g(-2), g(2), g(0) and g(3)
b) On what interval(s) is g increasing
c) What are the maximum and minimum values of g over [-2,3]?
x
9. a) Given
5 x 3  40   f (t )dt
c
i)
ii)
Find f(x).
Find the value of c
3
b) If F(x) =

1  t 16 dt , Find F ( x) .
x
x
10. Let
f(x) =
1
 1 t
4
dt for all real numbers x.
0
a)
b)
c)
d)
Find f(0)
Find f (1)
Justify that f(3) - f(1) < 1.
Justify that f(x) + f(-x) = 0 for all real numbers x.
2x
11. Let F ( x) 

t 2  t dt .
1
a) Find F (x ).
b) Find the domain of F.
c) Find
F ( x)
lim
x 12
e)
Find the length of the curve y = F(x) for [1,2]
x
12. Let f be a continuous function with domain x > 0 and let F be the function given by F ( x ) 
for x > 0. Suppose that F(ab) = F(a) + F(b) for all a > 0 and b > 0 and that F (1)  3.
a) Find f(1).
b) Prove that aF (ax)  F ( x) for every positive constant a.
c) Use the results from parts (a) and (b) to find f(x). Justify your answer.
 f (t )dt
1