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