Transcript Fundamental Theorem of Calculus

Fundamental Theorem of
Calculus
Section 4.4
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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)
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Examples
Example:

3
1
( x 2  4) dx 
Calculator check>> CALC 7:
 f (x)dx
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Examples
Example:
9
6
4
xdx 
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Examples
Example:

3
3
v 3 dv 
1
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Examples
Example:

4
1
t 2
t
dt 
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Examples
Example:

5
1
x  2 dx 

  x  2 , x  2
x2  

 x  2, x  2
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Examples
Find the area bounded by the graphs of y = x + sin x,

the x-axis, x = 0, and x =
2
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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
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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) 
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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.
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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
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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
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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
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Homework
Sect 4.4 page 291 #5 – 23 odd, 27, 29, 33 – 39 odd, 47
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