The Fundamental Theorem of Calculus

Inverse Operations

Fundamental Theorem of Calculus

Discovered independently by Gottfried Liebnitz and Isaac Newton

Informally states that differentiation and definite integration are inverse operations.

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

af x dx F b F a

Guidelines for Using the Fundamental Theorem of Calculus

1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum.

2. When applying the Fundamental Theorem of Calculus, the following notation is used

Guidelines

( ) ( )] ( ) ( )b b

aaf x dx F x F b F a

It is not necessary to include a constant of integration C in the antiderivative because they cancel out when you subtract.

Evaluating a Definite Integral

7 7 722 2

Evaluate each definite integral.

3dv 3 3 ] 21 6 15dv v

1

22

1( )u du

u

21 2 1 1

22

2 2

]2

1 21 1

2 1 2 2

1 32

2 2

uu u dx u

Evaluate the Definite Integral

24

20

1 sin

cos

2

4 420 0

cos1

cosd d

40] 0

4 4

Evaluate the Definite Integral

24

0sec x dx

4

0tan tan tan 0 1 0 1

4x

Definite Integral Involving Absolute Value Evaluate

2

02 1

The absolute value function has to be broken up

into its two parts:

12 1

22 11

2 12

x dx

x if xx

x if x

Definite Integral Involving Absolute Value

12

21

02

122 221

02

2 22

Now evaluate each part separately

2 1 (2 1)

1 1 1 10 2 2

2 2 2 2

1 1 52

4 4 2

x dx x dx

x x x x

Using the Fundamental Theorem to Find Area

Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the x-axis, and the vertical lines x = 0 and

x = 2

Using the Fundamental Theorem to Find Area

2 3

0

24 2

0

2 3 2

2 32

4 2

8 6 4 0 0 0

6

Area x x dx

x xx

The Mean Value Theorem for Integrals

If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that

( ) ( )( ).b

af x dx f c b a

Average Value of a Function

This is just another way to write the Mean Value Theorem (mean = average in mathematics)

If f is integrable on the closed interval

[a,b], then the average value of f on the interval is

Average Value of a Function

( )b

af x dx

cb a

Finding the Average Value of a Function

Find the average value of f(x) = sin x on the interval [0, ]

0 0sin cos

01 1 2

x dx xc

Force

The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is

[0, /3] and F(0) = 500.

Force

(a) Find F as a function of x.

F(x) = 500 sec2 x

(b) Find the average force exerted by the press over the interval [0, /3]

Force

32

0

30

500 sec

03

500 tan

3

500 tan tan 03

3

x dx

F

x

Force

500 3 0

3

500 3 1500 3827

3

Newtons

Second Fundamental Theorem of Calculus

If f is continuous on an open interval I containing a, then, for every x in the interval,

( ) ( )x

a

df t dt f x

dx

Using the Second Fundamental Theorem of Calculus

Evaluate

2

0( ) ( 1)

xdF x t t dt

dx

2 3( 1)x x x x

Second Fundamental Theorem of Calculus

Find F’(x) of

2

32

1( )

xF x dt

t

6 5

1 22x

x x