Fundamental Theorem of Calculus
Download
Report
Transcript Fundamental Theorem of Calculus
Fundamental Theorem of
Calculus
Section 4.4
1
Fundamental Theorem of Calculus
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
f ( x)dx F (b) F (a)
a
We express this as
f ( x)dx F ( x) a
b
a
b
F (b) F (a)
2
Examples
Example:
3
1
( x 2 4) dx
Calculator check>> CALC 7:
f (x)dx
3
Examples
Example:
9
6
4
xdx
4
Examples
Example:
3
3
v 3 dv
1
5
Examples
Example:
4
1
t 2
t
dt
6
Examples
Example:
5
1
x 2 dx
x 2 , x 2
x2
x 2, x 2
7
Examples
Find the area bounded by the graphs of y = x + sin x,
the x-axis, x = 0, and x =
2
8
Mean Value Theorem for Integrals
If f is continuous on [a, b], then there exists a number c
in the closed interval [a, b] such that
b
f ( x)dx
f (c) (b a)
a
Somewhere between the inscribed and circumscribed rectangles there is a
rectangle whose area is precisely equal to the area under the curve.
Inscribed Rectangle
Mean Value Rectangle
Circumscribed Rectangle
9
Average Value of a Function
The value of f(c) in the Mean Value Theorem for
Integrals is called the average value of f on the
interval [a, b].
b
Since
f ( x)dx
f (c) (b a) then solving for f(c) gives
a
f (c)
10
Example
Find the average value of f(x) = sin x on [0, ] and all
values of x for which the function equals its average
value.
11
The Second Fundamental Theorem
of Calculus
f
a
x
a
f (t )dt
x
Using x as the upper limit of integration.
d x
f ( x)
f
(
t
)
dt
dx a
12
Example
Integrate to find F as a function of x and demonstrate the Second
Fundamental Theorem of Calculus by differentiating the result.
x
1. F ( x) t dt
4
13
Example
Integrate to find F as a function of x and demonstrate the Second
Fundamental Theorem of Calculus by differentiating the result.
x
2. F ( x) sec t tan t dt
3
14
Homework
Sect 4.4 page 291 #5 – 23 odd, 27, 29, 33 – 39 odd, 47
15