The First Fundamental Theorem of Calculus

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