Section 5.3: Evaluating Definite Integrals Practice HW from Stewart Textbook (not to hand in) p. 374 # 1-27 odd, 31-43 odd.

Section 5.3: Evaluating Definite Integrals

Practice HW from Stewart Textbook (not to hand in)

p. 374 # 1-27 odd, 31-43 odd

Definite Integral

The definite integral is an integral of the form

This integral is read as the integral from a to b of . The numbers a and b are said to be the

limits of integration. For our problems, a < b.

Definite Integrals are evaluated using The Fundamental Theorem of Calculus.

dxxfb

a

)(

dxxf )(

Fundamental Theorem of Calculus

Let be a continuous function for

and be an antiderivative of . Then

)(xf bxa )(xF )(xf

)()()( )( aFbFxFdxxfba

b

a

Example 1: Evaluate .

Solution:

dxx 22

1

Example 2: Evaluate .

Solution:

dxxx )23(3

1

2

Additional Integration Formulas

1.

2.

dxe x

dxekx

dxa x

3.

4.

dxx

1

12

dxx

1

12

Example 3: Evaluate

Solution:

dxx

e x )1

1(

2

1

02

2

Example 4: Evaluate

Solution:

duu

u

43

13

2

Example 5: Evaluate

Solution:

dxx )1(3

1

23

Recall that if for . Then

Definite Integral:

0)( xf bxa

bxax

xfdxxf

b

a

for axis

theand )(Between Area )(

Example 6: Find the area under the graph of

on [0, 2].

Solution:

12 xy

Example 7: Evaluate

Solution:

2

1

2dxe x

Note

For a function f (x) that is both positive and negative

over an interval, the total area is the area enclosed by

the negative part of the curve minus the negative part

of the curve.

Example 8: Consider the function

over the interval . The graph of the function

over this interval is given by

xxxxf 34)( 23

30 x

Find the total area enclosed between the function and

the x axis.

Solution: