Section 4.2
10
Section 4.2 The Definite Integral
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Section 4.2. The Definite Integral. THE DEFINITE INTEGRAL. - PowerPoint PPT Presentation
Transcript of Section 4.2
Section 4.2
The Definite Integral
If f is a continuous function defined for a ≤ x ≤ b, we
divide the interval [a, b] into n subintervals of equal
width Δx = (b − a)/n. We let a = x0, x1, x2, x3, . . . ,
xn = b be the endpoints of these subintervals and we let
be any sample points in these
subintervals, so that lies in the ith subinterval
[xi − 1, xi]. Then the definite integral of f from a to b
is
THE DEFINITE INTEGRAL
**2
*1 ,, nxxx
*ix
b
a
n
ii
nxxfdxxf
1
*lim)(
The symbol ∫ is called an integral sign.
In the notation , f (x) is called the integrand, and a and b are called the limits of integration; a is the lower limit and b is the upper limit.
The symbol dx has no official meaning by itself;
is all one symbol.
The procedure of calculating an integral is called integration.
REMARKS
b
adxxf )(
b
adxxf )(
THE RIEMANN SUM
The sum
from the definition of the definite integral is called a Riemann sum after the German mathematician Bernhard Riemann.
n
ii xxf
1
*
If f (x) ≥ 0 (that is, the graph lies above the x-
axis), gives the area under the curve.
If the graph of f (x) lies both above and below the x-axis, then the definite integral gives the net area (the area above the x-axis subtracted by the area below the x-axis); that is
NET AREA
b
adxxf )(
b
aAAdxxf downup)(
If we are approximating a definite integral, it is often better to let be the midpoint of the ith subinterval. This results in the Midpoint Rule.
THE MIDPOINT RULE
*ix
iiiii
n
n
ii
b
a
xxxxx
n
abx
xfxfxxxfdxxf
,ofmidpointand
where
)()()()(
1121
11
BASIC PROPERTIES OF THE DEFINITE INTEGRAL
a
a
a
b
b
a
dxxf
dxxfdxxf
0)(
)()(
CONSTANT MULTIPLE AND ADDITION/SUBTRACTION PROPERTIES
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
dxxgdxxfdxxgxf
dxxfcdxxfc
dxxgdxxfdxxgxf
abcdxc
)()()()(.4
)()(.3
)()()()(.2
)(.1
INTERVAL ADDITIVITY PROPERTY
b
a
b
c
c
adxxfdxxfdxxf )()()(.5
6. If f (x) ≥ 0 for a ≤ x ≤ b, then
7. If f (x) ≥ g(x) for a ≤ x ≤ b, then
8. If m ≤ f (x) ≤ M for a ≤ x ≤ b, then
COMPARISON PROPERTIES
b
adxxf .0)(
.)()( b
a
b
adxxgdxxf
.)()()( abMdxxfabmb
a
