Math 1B
§ 5.3 The Fundamental Theorem of Calculus
Overview: Here we are introduced to the Fundamental Theorem of Calculus, which is the central theorem of
integral calculus. It connects integration and differentiation, allowing us to compute integrals using an
antiderivative of the integrand function rather than by taking limits of Riemann sums as we did in Section 5.2. It
was the Englishman Issac Barrow (1630 – 1677) who first noticed this connection and then it was Leibniz (1646
– 1716) and Newton (1642 – 1727) who took advantage of this relationship between integration and
differentiation and started mathematical developments that fueled the scientific revolution for the next 200
years.
The Fundamental Theorem of Calculus - Part I
Consider the integral
x
∫ f (t )dt , where
a
y = f (t) .
Since the upper limit is a variable, this actually defines
a function, g( x ) . For each value of the input x, there is
€
€
a well-defined numerical output, in this case the definite
integral of f from a to x, which means the area under the curve y = f ( t ) from a to x. If f is any
continuous function, then the Fundamental Theorem asserts that g is a differentiable function of x
€
whose derivative is f itself. That is, at every value of x,
d
d
g( x ) =
dx
dx
x
€
∫ f (t)dt = f ( x )
a
Example: Using the graph of€ y = f (t ) (shown to the left below) and g( x ) =
€
x
∫ f (t)dt , which means
0
g( x ) is the area under f ( t ) from 0 to x, find g(2),g( 4 ),g(6),g( 7),g(10),g(12), and g(14 ) , and then
sketch a rough graph of g. Make a note of when g is increasing and decreasing, and also any local
maxima or minima. Do you see a relationship between f and g?
€
34
€
€
€
x
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
-2 -1-2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
y
Stewart – 7e
1
If we assume that f ( t ) ≥ 0 and define the area under y = f ( t )
from a to x as g( x ) =
x
∫ f (t)dt
–
g( x +
€h )
(area from a → x+h)
€
€
then
a
g( x )
€
≈
hf ( x )
(area from a → x)
So, dividing both sides
€ by h we get
(area from x → x+h)
g( x + h ) − g( x )
≈ f ( x)
€
h
g ( x + h) − g ( x )
Now if we take the limit as h → 0 we get lim
= f ( x)
h→0
h
€


g ′( x )
So
d
d
g ( x) =
dx
dx
x
∫ f (t ) dt = f ( x ) , which leads us to
a
...
The Fundamental Theorem of Calculus – Part 1 (FTC1): If f is continuous on [ a,b] , then the
function g defined by g( x ) =
x
∫ f (t)dt , a ≤ x ≤ b , is continuous on [a,b] and differentiable on (a,b) ,
a
and g"( x ) = f ( x ) .
€
Example: Use Part 1 of the Fundamental
Theorem of Calculus
to find the derivative of
€
€
€ each function.
€
€
a) g( x ) =
∫
x
5
cos tdt
€
3 t 2 −t
b) h ( x ) =
∫
d) g( x ) =
∫ (1+ v )
x
e
dt
€
c) y =
∫
x2 3
0
t dt
cos x
1
2 10
dv
€
Stewart – 7e
2
The Fundamental Theorem of Calculus – Part II
In Section 5.2, we computed integrals by taking the limit of a Riemann Sum. This is a long
procedure and is sometimes very difficult. The second part of the Fundamental Theorem of
Calculus gives us a much simpler method of computation of definite integrals.
The Fundamental Theorem of Calculus – Part 2 (FTC2): If f is continuous on [ a,b] , then
b
b
a
a
∫ f ( x )dx = F (b) − F (a) or F ( x )]
where F is any antiderivative of f, that is, a function such that
F" = f .
€
FTC2 tells us that €
if we know the antiderivative F of f, then we can evaluate
€
€
b
∫ f ( x )dx simply by
a
subtracting the values of F at the endpoints of the interval [a, b].
For example, if v ( t ) is the velocity of an object and s( t ) is its position at time t, then v ( t ) = s"( t ) , so s
€
is an antiderivative of v. Now, if we assume that the object always moves in the positive direction, as
we did in 5.1, then the area under the velocity curve is equal to the distance traveled, which is
b
∫ a v (t€)dt = s(b) − s(a) .
€
€
Recall the antiderivative formulas from 4.9:
€
Function
Antiderivative
Function
Antiderivative
c f (x)
c F(x) + C
sin x
−cos x + C
f ( x ) + g( x )
F ( x ) + G( x ) + C
s e c2 x
tan x + C
xn ( n ≠ − 1)
x n +1
+C
n +1
sec x tan x
sec x + C
1
x
ln x + C
ex
ex + C
1
1 + x2
t a n− 1 x + C
cos x
sin x + C
k
kx + C
1
1 − x2
s i n− 1 x + C
Example: Evaluate the following integrals, if possible.
a)
∫
8
3
1
x dx
b)
2
∫ ( y −1) (2y +1) dy
0
€
Stewart – 7e
3
c)
∫
4
dt
0 t2 +1
1
d)
∫
f)
∫
2
1
4 + w2
dw
w3
€
€
e)
2π
∫ (1+ cosθ )dθ
π
€
2
−1
4
dx
x3
€
Differentiation and Integration as Inverse Processes
The Fundamental Theorem of Calculus (FTC) tells us that differentiation and integration are
inverse processes. This means when you take an integral, you can always check your answer by taking
the derivative to see if you get your original function back again. The FTC provides a systematic
approach to some otherwise extremely difficult problems.
The Fundamental Theorem of Calculus – Parts 1 and 2:
Suppose f is a continuous function on [a, b].
x
1. If g( x ) = ∫ a f ( t ) dt , then g"( x ) = f ( x ) .
(This says that if f is integrated and then the result is differentiated, we arrive back at the
original function f.)
€
€
2. If F is any antiderivative of f, such that F " = f , then
b
∫ f ( x )dx = F (b) − F (a) .
a
(This says that if we take a function F, first differentiate it, getting f, and then integrate the
result on the interval [a, b], we arrive back at the original function F, but in the form
F (b) − F ( a) .)
€
€
Stewart – 7e
€
4