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The First Fundamental Theorem of Calculus. First FTOC. ** Notice!! You do not have to include a “C” when you integrate f(x). Example Evaluating an Integral. Example. The Derivative of an Integral. Now let’s see some more examples / practice of definite integral problems. - PowerPoint PPT Presentation
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The First Fundamental Theorem of Calculus
If is continuous at every point of [ , ], and if is any antiderivative
of on [ , ], then ( ) ( ) - ( ).
This part of the Fundamental Theorem is also called the
.
b
a
f a b F
f a b f x dx F b F a Integral
Evaluation Theorem
( ) ( ) ( )
Any definite integral of any continuous function can be calculated without
taking limits, without calculating Riemann sums, and often without effort -
so long as an antiderivative
b
a f x dx F b F a
f
of can be found.f
First FTOC
b
aaFbFdxxf )()()(
** Notice!! You do not have to include a “C” when you integrate f(x)
Example Evaluating an Integral
3 2
-1Evaluate 3 1 using an antiderivative.x dx
33 2 3
-1 1
33
3 1
3 3 1 1
32
x dx x x
3
1
3 )1( dxx
24
14
13
4
81
4
3
1
4
xx
Example
2
)1()1(
)0cos(cos
cos
sin
0
0
x
dxxFind
The Derivative of an Integral
( ) ( ).x
a
df t dt f x
dx
Now let’s see some more examples / practice of definite
integral problems
The Mean Value Theorem for Definite Integrals
If is continuous on [ , ], then at some point in [ , ],
1( ) ( ) .
( ) ( )( )
b
a
b
a
f a b c a b
f c f x dxb a
f x dx f c b a
The Mean Value Theorem for Definite Integrals
The mean value theorem for definite integrals says that for a continuous function, at some point on the interval the actual value will equal the average value.
Mean Value Theorem (for definite integrals)
If f is continuous on then at some point c in , ,a b ,a b
1
b
af c f x dx
b a
p
Average(Mean) Value
b
adxxf
abfav )(
1)(
Example Applying the Definition
2Find the average value of ( ) 2 on [0,4].f x x
4 2
0
1( ) ( )
1 2 Use NINT to evaluate the integral.
4 01 40
4 3
10
3
b
aavg f f x dxb a
x dx
The average value of a function is the value that would give the same area if the function was a constant:
21
2y x
3 2
0
1
2A x dx
33
0
1
6x
27
6
9
2 4.5
4.5Average Value 1.5
3
Area 1Average Value
Width
b
af x dx
b a
1.5
Now let’s see some examples using the Mean Value Theorem
for Integrals and to find the average value of a function
The Second Fundamental Theorem of Calculus
( ) ( )
Every continuous function is the derivative of some other function.
Every continuous function has an antiderivative.
The processes of integration and differentiation are inverses of o
x
a
df t dt f x
dx
f
ne another.
The Second Fundamental Theorem of Calculus
If is continuous on [ , ], then the function ( ) ( )
has a derivative at every point in [ , ], and
( ) ( ).
x
a
x
a
f a b F x f t dt
x a b
dF df t dt f x
dt dx
The Derivative of an Integral
( ) ( ).x
a
df t dt f x
dx
x
a
df t dt f x
dx
Second Fundamental Theorem:
1. Derivative of an integral.
a
xdf t dt
xf x
d
2. Derivative matches upper limit of integration.
Second Fundamental Theorem:
1. Derivative of an integral.
a
xdf t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
Scond Fundamental Theorem:
x
a
df t dt f x
dx
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
New variable.
Second Fundamental Theorem:
Example
22 1
1
1
1
xdt
tdx
d x
Example Applying the Fundamental Theorem
Find sin .xdtdt
dx
sin sinxdtdt x
dx
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