Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:

Definition:

the definite integral of f from a to b is

provided that this limit exists. If it does exist, we say that is f integrable on [a,b]

Sec 5.2: THE DEFINITE INTEGRAL

xxfdxxfn

ii

n

b

a

1

*)(lim)(

n

ii

nn

nxxfRA

1

)(limlim

Note 1:


b

adxxf )(

Integral sign

limits of integration

lower limit aupper limit b

integrand

The procedure of calculating an integral is called integration.

The dx simply indicates that the independent variable is x.

Note 2:


numberdxxfb

a )(

x is a dummy variable. We could use any variable

b

a

b

a

b

adzzfdttfdxxf )()()(

Note 3:


Riemann sum

n

ii

nn

nxxfRA

1

*)(limlim

Riemann sum is the sum of areas of rectangles.

0)( xf

Note 4:


Riemann sum is the sum of areas of rectangles.

0)( xf

0)( xf

area under the curveb

adxxf )(

Note 5:


If takes on both positive and negative values,

the Riemann sum is the sum of the areas of the rectangles that lie above the -axis and the negatives of the areas of the rectangles that lie below the -axis (the areas of the gold rectangles minus the areas of the blue rectangles).

A definite integral can be interpreted as a net area, that is, a difference of areas:

where is the area of the region above the x-axis and below the graph of f , and is the area of the region below the x-axis and above the graph of f.

21)( AAdxxfb

a

Note 6:


not all functions are integrable

n

ii

nn

nxxfRA

1

)(limlim

f(x) is cont [a,b] integrable )(xf exist )(b

adxxf

f(x) has only finite number of removable discontinuities

integrable )(xf exist )(b

adxxf

f(x) has only finite number of jump discontinuities


adxxf


4

4)( dxxf

4

4)( dxxf

4

4)( dxxf

f(x) is cont [a,b] integrable )(xf exist )(b

adxxf

f(x) has only finite number of removable discontinuities


adxxf

f(x) has only finite number of jump discontinuities


adxxf

Note 7:


the limit in Definition 2 exists and gives the same value no matter how we choose the sample points

n

ii

nn

nxxfSA

1

*)(limlim



Term-092

Example:


(a) Evaluate the Riemann sum for taking the sample points to be

right endpoints and a =0, b =3, and n = 6.

xxxf 6)( 3

(b) Evaluate 3

0

3 6 dxxx

Example:Example:


(a) Set up an expression for

as a limit of sums

3

1dxex

Example:Evaluate the following integrals by interpreting each in terms of areas.

1

0

21) dxxa

3

0)1() dxxb

Evaluate the following integrals by interpreting each in terms of areas.


Midpoint Rule


We often choose the sample point to be the right endpoint of the i-th subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose to be the midpoint of the interval, which we denote by .

*ix

21 ii

i

xxx


Property (1)


b

a

a

bdxxfdxxf )()(

Example:

0 2 cos

dxxx

0

2 cos dxxx


Property (2)

0)( a

adxxf


Property (3)

b

c

c

a

b

adxxfdxxfdxxf )()()(

Example:


Note: Property 1 says that the integral of a constant function is the constant times the length of the interval.

Use the properties of integrals to evaluate

1

0

2 )34( dxx


Term-091



Term-092

Example:


Use Property 8 to estimate

1

0

2

dxe x


SYMMETRY

Suppose f is continuous on [-a, a] and even

a

adxxfdxxf

0

0)()(

Suppose f is continuous on [-a, a] and odd

0)( a

adxxf

)()( xfxf

)()( xfxf

aa

adxxfdxxf

0)(2)(

a

adxxfdxxf

0

0)()(


Term-102


Term-102


Term-091


Term-091


n

ii

nn

nxxfRA

1

)(limlim


Term-102


Term-082


Term-082


Term-103


Term-103


Term-103


Term-103


Term-092


Term-082

Documents

Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2: