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Riemann Sum. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum . The width of a rectangle is called a subinterval . The entire interval is called the partition . subinterval. partition. Subintervals do not all have to be the same size. - PowerPoint PPT Presentation
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Riemann Sum
k
n
kk xcf
1
When we find the area under a curve by adding rectangles, the answer is called a Rieman sum.
21 18
V t
subinterval
partition
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
Subintervals do not all have to be the same size.
21 18
V t
subinterval
partition
If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by . P
As gets smaller, the approximation for the area gets better.
P
0 1
Area limn
k kP k
f c x
if P is a partition of the interval ,a b
0 1
limn
k kP k
f c x
is called the definite integral of
over .f ,a b
If we use subintervals of equal length, then the length of a
subinterval is:b axn
The definite integral is then given by:
1
limn
kn k
f c x
1
limn
kn k
f c x
Leibnitz introduced a simpler notation for the definite integral:
1
limn b
k an k
f c x f x dx
Note that the very small change in x becomes dx.
Limit of Riemann Sum = Definite Integral
dxxfxcfb
ak
n
kkP
10
lim
dxxfxcfb
ak
n
kkn
1
lim
nabxIf
b
af x dx
IntegrationSymbol
lower limit of integration
upper limit of integration
integrandvariable of integration(dummy variable)
It is called a dummy variable because the answer does not depend on the variable chosen.
The Definite Integral
( )b
a f x dx
Existence of Definite Integrals
All continuous functions are integrable.
Example Using the Notation
th
2
1
The interval [-2,4] is partitioned into subintervals of equal length 6 / .Let denote the midpoint of the subinterval. Express the limit
lim 3 2 5 as an integral.k
n
k kn k
n x nm k
m m x
2 4 2
21
lim 3 2 5 3 2 5n
k kn km m x x x dx
Area Under a Curve (as a Definite Integral)
If ( ) is nonnegative and integrable over a closed interval [ , ],then the area under the curve ( ) from to is the
, ( ) .b
a
y f x a by f x a b
A f x dx
integral
of from to f a b
Note: A definite integral can be positive, negative or zero, but for a definite integral to be interpreted as an area the function MUST be continuous and nonnegative on [a, b].
Area
Area= ( ) when ( ) 0.
( ) area above the -axis area below the -axis .
b
a
b
a
f x dx f x
f x dx x x
Integrals on a Calculator
b
abaxxffnIntdxxf ),,),(()(
Example Using NINT
2
-1Evaluate numerically. sinx xdx
NINT( sin , , -1,2) 2.04x x x
Properties of Definite Integrals
Order of Integration
b
a
a
bdxxfdxxf )()(
Zero
a
adxxf 0)(
Constant Multiple
b
a
b
a
b
a
b
a
dxxfdxxf
dxxfkdxxfk
)()(
)()(
Sum and Difference
b
a
b
a
b
adxxgdxxfdxxgxf )()()()(
Additivity
b
a
c
b
c
adxxfdxxfdxxf )()()(
1. If f is integrable and nonnegative on the closed interval [a, b], then
2. If f and g are integrable on the closed interval [a, b] and for every x in [a, b], then
0 ( )b
af x dx
( ) ( )f x g x( ) ( )
b b
a af x dx g x dx
Preservation of Inequality