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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:
Sec 5.2: THE DEFINITE INTEGRAL
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:
Sec 5.2: THE DEFINITE INTEGRAL
numberdxxfb
a )(
x is a dummy variable. We could use any variable
b
a
b
a
b
adzzfdttfdxxf )()()(
Note 3:
Sec 5.2: THE DEFINITE INTEGRAL
Riemann sum
n
ii
nn
nxxfRA
1
*)(limlim
Riemann sum is the sum of areas of rectangles.
0)( xf
Note 4:
Sec 5.2: THE DEFINITE INTEGRAL
Riemann sum is the sum of areas of rectangles.
0)( xf
0)( xf
area under the curveb
adxxf )(
Note 5:
Sec 5.2: THE DEFINITE INTEGRAL
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:
Sec 5.2: THE DEFINITE INTEGRAL
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
integrable )(xf exist )(b
adxxf
Sec 5.2: THE DEFINITE INTEGRAL
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
integrable )(xf exist )(b
adxxf
f(x) has only finite number of jump discontinuities
integrable )(xf exist )(b
adxxf
Note 7:
Sec 5.2: THE DEFINITE INTEGRAL
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
Example:
Sec 5.2: THE DEFINITE INTEGRAL
(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:
Sec 5.2: THE DEFINITE INTEGRAL
(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
Sec 5.2: THE DEFINITE INTEGRAL
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
Example:
Sec 5.2: THE DEFINITE INTEGRAL
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
Sec 5.2: THE DEFINITE INTEGRAL
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)()(