-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
1/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
1
Mean Value Theorem for Definite Integrals
y =f(x) is a positive continuous faction over[a, b].
Geometrically, the Mean Value
Theorem says that there is a
number c in [a, b] such that the
rectangle with height equal to
the average value (c) of the
function and base width b-a has
exactly the same area as the
region beneath the graph of
from a to b.
The Mean Value Theorem for Definite Integrals asserts (claims)
that this average value is alwaystaken on at least once by the
function in the interval.
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
2/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
2
Example
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
3/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
3
Fundamental Theorem, Part 1
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
4/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
4
Example
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
5/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
5
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
6/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
6
Example
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
7/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
7
Example
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
8/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
8
Example
Solution
The integral value = 0, but is the area under the curve is also = 0?
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
9/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
9
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
10/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
10
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
11/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
11
Solution
=
+
=
+
= , ,
=
+
+
=
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
12/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
12
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
13/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
13
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
14/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
14
Because of symmetry about the y-axis ,
Solution
-
7/29/2019 5.4 the Fundamental Theorem of Calculus 2013
15/15
Chapter 5: Integration 5.4 The Fundamental Theorem of Calculus
15
Solve problems
1 to 46
Solution
The area of the rectanglebounded by the lines:
Rectangle area :
