Transcript POWER SERIES - MATHCHICK.NET

THE FUNDAMENTAL
THEOREM OF CALCULUS
Section 4.4
When you are done with your
homework, you should be able
to…
– Evaluate a definite integral using the
Fundamental Theorem of Calculus
– Understand and use the Mean Value
Theorem for Integrals
– Find the average value of a function
over a closed interval
– Understand and use the Second
Fundamental Theorem of Calculus
•
Galileo lived in Italy from 1570-1642. He
defined science as the quantitative description
of nature—the study of time, distance and
mass. He invented the 1st accurate clock and
telescope. Name one of his advances.
A. He discovered laws of motion for a falling
object.
B. He defined science.
C. He formulated the language of physics..
D. All of the above.
THE FUNDAMENTAL
THEOREM OF CALCULUS
• Informally, the theorem states that
differentiation and definite integrals
are inverse operations
• The slope of the tangent line was defined
using the quotient dy
dx
• The area of a region under a curve was
defined using the product dydx
– The Fundamental Theorem of Calculus states
that the limit processes used to define the
derivative and definite integral preserve this
relationship
x
x
x
y
y
y
Seca nt
l ine
Slope 
y
x
Tan gent
l ine
Slope 
y
x
x
y
Area of
Rectangl e
Area  yx
Area of
Reg ion
unde r
curve
Area  yx
Theorem: The Fundamental
Theorem of Calculus
• If a function f is continuous on the
closed interval  a, b and F is an
antiderivative of f on the interval  a, b,
then b

a
f  x  dx  F  b   F  a 
Guidelines for Using the Fundamental Theorem
of Calculus
1. Provided you can find an antiderivative of f, you
now have a way to evaluate a definite integral
without having to use the limit of a sum.
2. When applying the Fundamental Theorem of
Calculus, the following notation is convenient:
b
 f  x  dx  F  x 
b
a
a
 F b  F  a 
3. It is not necessary to include a constant of
integration in the antiderivative because
b

f  x  dx  F  x   C  a
b
a
  F  b   C    F  a   C 
 F b  F  a 
Example: Find the area of the
region bounded by the graph of
2
y  x  3 , the x-axis, and the
vertical linesx  1 and x  3 .
hx = x2+3
q  y = 1
ry  = 3
15
10
5
-4
-2
2
4
Find the area under the
curve bounded by the
3
graph of f  x   x  2, x  1,
and the x-axis and the
y-axis.
9/4
0.0
4
f x = -x3+2
gy = -1
2
hy = 0
-5
5
THE MEAN VALUE THEOREM
FOR INTEGRALS
If f is continuous on the closed
interval  a, b, then there exists a
number c in the closed interval  a, b
such that
b

a
•
f  x  dx  f  c  b  a 
So…what does this mean?!
Somewhere between the inscribed and circumscribed
rectangles there is a rectangle whose area is
precisely equal to the area of the region under the
curve.
f c 
a
c
Me an Value Rectangl e
b
AVERAGE VALUE OF A
FUNCTION
• If f is integrable on the closed
interval  a, b, then the average value
of f on the interval is
b
1
f
x
dx



ba a
Find the average value of
the function
3
3x
dx

2
13.0
0.0
1
THE SECOND FUNDAMENTAL
THEOREM OF CALCULUS
If f is continuous on an open
interval I containing c,
then, for every x in the
interval,

d 
  f  t  dt   f  x 
dx  a

x