5.4 THE FUNDAMENTAL THEOREM OF
CALCULUS
THE FUNDAMENTAL THEOREM OF CALCULUS
(Part 2 of the FTC in your book)
 If f is continuous on [a, b] and F is an
antiderivative of f on [a, b], then

b
 f ( x)dx  F (b)  F (a)
a
**F(b) – F(a) is often denoted as
F ( x)
b
a
This part of the FTC is significant because it allows us to
evaluate definite integrals.
FTC…THE PART THAT CONNECTS IT ALL…
x

If f is continuous on [a, b] and F ( x) 
 f (t )dt
a
then F’(x) = f(x) at every point x in [a, b].
You may also see this as
x
dF d
d
  f (t )dt  f ( x)
dx dx a
dx
WHAT’S THE SIGNIFICANCE?

Every continuous f is thex derivative of some
other function, namely  f (t )dt
a

Every continuous function has an
antiderivative.

The processes of integration and differentiation
are inverses of each other!
EXAMPLE

If
x
F ( x)   (t 2  4)dt ,
integrate to find F(x). Then,
1
differentiate to find F’(x).
x
1 3

F ( x)   t  4t 
3
1
1
 1

  x3  4 x     4 
3
 3


1 3
11
x  4x 
3
3
F ' ( x)  x  4
2
EXAMPLE

Using FTC for
x
F ( x)   (t 2  4)dt ,
1
F ' ( x)  x  4
2
find F’(x).
EXAMPLE

If
x
h( x)   t  1dt
, find h’(8).
3
h' ( x )  x  1
h' (8)  8  1  3
EXAMPLE
1
1
dy
 Find
if y   3 dt
t
dx
2x
2x
1
y    3 dt
t
1
1

2
3
(2 x)
1
 3
4x
(change bounds)
EXAMPLE

Construct a function in the form of
x
y   f (t )dt  C
a
that has tan x as the derivative and satisfies
f(3) = 5.
x
dy
tan tdt  tan x and

dx 3
x
x
 tan tdt  0
3
y   tan tdt  5
3
when x = 3, so
TOTAL AREA VS. NET AREA

Net area counts area below the x-axis as
negative area.
 Computing
area on an entire interval using
antiderivatives helps us find net area

Total area is the entire amount of area
enclosed by a graph
 Problems
that say find “area” from here on out
imply for you to find “total area.”
FINDING TOTAL AREA ANALYTICALLY
To find the area between the graph of y = f(x) and
the x-axis over the interval [a, b] analytically,
1. Break [a, b] apart using the zeros of f
2. Integrate f over each subinterval
3. Add the absolute values of the integrals
EXAMPLE

Find the area of the region between the curve
y = 4 – x2 and the x-axis over [0, 3].
1. Break the curve into two subintervals since part of it is
above the x-axis and the other is below.
2. Break the integral into [0, 2] and [2, 3]
2
3
2
2
(
4

x
)
dx

(
4

x
)dx


0
(subtract to make the
second area positive)
2
2
3

x  
x 
4 x    4 x  
3 0 
3 2

3
3
16  7 23
 

3
3
3
EXAMPLE

Find the net area of the region between the
curve y = 4 – x2 and the x-axis over [0, 3].
3
2
(
4

x
)dx

0
3

x 
4 x    12  9  3
3 0

3
TOTAL AREA USING FINT

To find the area between the graph of y = f(x)
and the x-axis over the interval [a, b]
numerically, evaluate
b

a
f ( x) dx