Download The Fundamental Theorems of Calculus

The Fundamental Theorems
of Calculus
Lesson 5.4
First Fundamental Theorem of Calculus
• Given f is
 continuous on interval [a, b]
 F is any function that satisfies F’(x) = f(x)
• Then

b
a
f ( x)dx  F (b)  F (a)
First Fundamental Theorem of Calculus
• The definite integral

b
a
f ( x)dx
can be computed by
 finding an antiderivative F on interval [a,b]
 evaluating at limits a and b and subtracting
• Try

7
3
6x dx
Area Under a Curve
• Consider
 
y  sin x  cos x on 0, 
 2
• Area =


2
0
sin x  cos x dx
Area Under a Curve
• Find the area under the following function on
the interval [1, 4]
y  ( x  x  1) x
2
Second Fundamental Theorem of
Calculus
• Often useful to think of the following form

x
a
f (t )dt
• We can consider this to be a function in
terms of x
View
Geogebra
Demo
x
F ( x)   f (t )dt
a
View QuickTime
Movie
Second Fundamental Theorem of
Calculus
• Suppose we are
given G(x)
x
G( x)   (3t  5)dt
4
• What is G’(x)?
Second Fundamental Theorem of
Calculus
• Note that
x
F ( x)   f (t )dt
a
Since this is a
constant …
• Then
• What about
 F ( x)  F (a )
d
 F ( x)  F (a )   f ( x)
dx
a
F ( x)   f (t )dt
x
?
Second Fundamental Theorem of
Calculus
• Try this
1
dt
x
1  3t
F ( x)  
2
dt
a
F ( x)   f (t )dt
x
 F (a )  F ( x)
so F '( x)   f ( x)
Assignment
• Lesson 5.4
• Page 329
• Exercises 1 – 49 odd