RIEMANN SUMS, THE DEFINITE

INTEGRAL, INTEGRAL AS

AREASection 5.2a

First, we need a reminder of sigma notation:

1

n

kk

a 1 2 3 1n na a a a a

How do we evaluate:

…and what happens if an “infinity” symbol appearsabove the sigma???

The terms go on indefinitely!!!

LRAM, MRAM, and RRAM are all examples ofRiemann sums, because of how they were constructed.

In this section, we start with a more general accountof these sums………………….observe…………….

We start with an arbitrary function f(x), defined on a closedinterval [a, b].

a b

Partition the interval [a, b] into n subintervals by choosing n – 1points between a and b, subject only to

1 2 1na x x x b

Letting a = x and b = x , we have apartition of [a, b]:

0 1 2, , , , nP x x x x

0 n

We start with an arbitrary function f(x), defined on a closedinterval [a, b].

The partition P determines n closed subintervals.thkThe subinterval is , which has length

1k k kx x x 1,k kx x

x

2x1x

1x0x a 1kx 2x

kx

kx 1nx nx b

nx

In each subinterval we choose some number, denoting thenumber chosen from the subinterval by .thk kc

On each subinterval, we create a rectangle that reaches fromthe x-axis to touch the curve at . ,k kc f c

a b

On each subinterval, we create a rectangle that reaches fromthe x-axis to touch the curve at . ,k kc f c

1c 2ckc nc

1 1,c f c

,n nc f c

2 2,c f c

,k kc f c

On each subinterval, we form theproduct

(which can be positive,negative, or zero…)

kf c x

Area of each rectangle!!!

a b1c 2c

kc nc

1 1,c f c

,n nc f c

2 2,c f c

,k kc f c

Finally, take the sum of these products:

1

n

n k kk

S f c x

This sum is called theRiemann sum for f on the interval [a, b]

As with LRAM, MRAM, and RRAM, all Riemann sums for agiven interval [a, b] will converge to common value, as long asthe subinterval lengths all tend to zero.

To ensure this last condition, we require that the longestsubinterval (called the norm of the partition, denoted ||P||)tends to zero…

Riemann Sums

Definition: The Definite Integral as aLimit of Riemann SumsLet f be a function defined on a closed interval [a, b]. For anypartition P of [a, b], let numbers be chosen arbitrarily in thesubintervals . 1,k kx x

kc

If there exists a number I such that

0 1

limn

k kP k

f c x I

no matter how P and the ‘s are chosen, then f is integrableon [a, b] and I is the definite integral of f over [a, b].

kc

Theorem: The Existence of DefiniteIntegralsIn particular, if f is continuous, then choices about partitions and ‘s don’t matter, as long as the longest subinterval tends tozero:kc

All continuous functions are integrable. That is, if afunction f is continuous on an interval [a, b], then itsdefinite integral over [a, b] exists.

This theorem allows for a simpler definition of the definite integralfor continuous functions. We need only consider the limit ofregular partitions (in which all subintervals have the samelength)…

The Definite Integral of a ContinuousFunction on [a, b]Let f be continuous on [a, b], and let [a, b] be partitioned into nsubintervals of equal length . Then thedefinite integral of f over [a, b] is given by

x b a n

1

limn

kn k

f c x

where each is chosen arbitrarily in the subinterval.kc thk

Integral NotationThe Greek “S” is changed to an elongated Roman “S,”so that the integral retains its identity as a “sum.”

1

limn

kn k

f c x

b

a

f x dxThis is read as “the integral from a to b of f of x dee x”

or “the integral from a to b of f of x with respect to x”

Integral Notation

b

a

f x dx

The function isthe integrand

x is the variableof integration (also

called a dummyvariable)

Upper limitof integration

Lower limitof integration

IntegralSign

Integral of f from a to b

When you find the valueof the integral, you haveevaluated the integral

A Quick Practice Problem

4x n The interval [–1, 3] is partitioned into n subintervals of equallength . Let denote the midpoint of thesubinterval. Express the given limit as an integral.

km thk

2

1

lim 3 2 5n

k kn k

m m x

23 2 5f x x x The function being integrated is

over the interval [–1, 3]...

3

2

1

3 2 5x x dx

Definition: Area Under a Curve (as aDefinite IntegralIf is nonnegative and integrable overa closed interval [a, b], then the area underthe curve of from a to b is theintegral of from a to b,

b

a

A f x dx

y f x

y f xf

Practice ProblemEvaluate the integral

2 2

24 x dx

What is the graph of the integrand???

2 2

24 2x dx

212r

From Geometry-Land:

Area = 21 2 22

(2, 0)(–2, 0)

(0, 2)

What happens when the curve isbelow the x-axis?

The area is negative!!!

b

a

f x dx Area = 0f x when

If an integrable function y = f (x) has both positive and negativevalues on the interval [a, b], add the areas of the rectanglesabove the x-axis, and subtract those below the x-axis:

For any integrable function,

b

af x dx = (area above x-axis) – (area below x-axis)

What happens with constantfunctions?

If f (x) = c, where c is a constant, on theinterval [a, b], then

b

af x dx

b

ac dx c b a

Does this make sense graphically???

3

28 dx

Quick Example:

8 3 2 40

Practice ProblemsUse the graph of the integrand and areas to evaluatethe given integral.

3 2

1 22 4x dx

1 1 3 12

2

Practice ProblemsUse the graph of the integrand and areas to evaluatethe given integral.

1

11 x dx

1 2 12

1

Practice ProblemsUse the graph of the integrand and areas to evaluatethe given integral.

3b

atdt

1 3 32b a b a

2 232b a

a b

3a

3b