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Module 4 : The Defnite Integral 

 This module will help you:

•  defne and interpret defnite integral

• identiy and distinguish the dierent properties o the defnite integral.

• evaluate defnite integrals. _________________________________________________________________________________________________ 

Definition:

If the function, is a continuous function in the closed interval from , , and is

any indefinite integral of then the definite integral of is given by:

 

where

The definite integral link the concept of area to other important concepts such as length, volume,decsity, probability, and other work.

Properties of the Definite Integral:

If and are continuous function on the interval of integration

1.

2.

3. for any constant,

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4.

5. , when

6. The First Mean Value Theorem:

for at least one value between and

7. If , then

To obtain the definite integral of a function, evaluate first its indefinite integral. Then applying

the limits of integration, that is, substitute the upper limit of integration to all the variables

contained in the indefinite integral, minus the function value of the indefinite integral using the lower

limit of integration.

EXAMPE 5:

1. valuate:

2. valuate:

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3. valuate:

!y simple substitution:

"et

4. valuate:

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5. valuate:

#sing integration by parts: "et $

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6. valuate:

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7. valuate:

!y substitution: "et

!. valuate:

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$ From

". valuate:

!y simple substitution: "et

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1#. valuate:

!y simple substitution: "et $ if

if

EXE$%I&E 4:

valuate the following integrals:

1.

2.

3.

4.

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5.

6.

7.

!.

".

1#.

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