Definite Integrals Finney Chapter 6.2 - PowerPoint Presentation

Slide1

Definite Integrals

Finney Chapter 6.2Slide2

Riemann Sums

The sums you studied in the last section are called

Riemann Sums

When studying area under a curve, we consider only intervals over which the function has positive values because area must be positiveRiemann Sums are more general and we may use any interval over which a function is continuousIntervals over which a function has negative values give the negative value of the area under the curve for that intervalWhat follows is a description of how a Riemann sum is created

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Riemann Sums

Creating a Riemann Sum is similar to what you practiced in the previous section

For a function

over a continuous interval

, we divide the interval into

subintervals by choosing points between and , such thatWe can think of this as a set, that is called a partition of

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Riemann Sums

The

subintervals are indexed by the right endpoint; that is, the first subinterval is

, the second subinterval is

, and the

th

subinterval is In a Riemann Sum, each subinterval can have any widthHence, the first subinterval has width , the second subinterval has width , the th subinterval has width (again, these values need not be equal)A value is selected from anywhere in each subinterval; the value from the th subinterval is denoted ; that is, is contained in  

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Riemann Sums

At each subinterval we construct a rectangle such that the height of the rectangle is

and the width is

The area of each such rectangle is

; this value can be positive, negative, or zero

Finally we take the sum of these products

The value of the sum depends on the partition and the choice of numbers and is called the Riemann Sum for on  5Slide6

Riemann Sums

Since the subintervals may have different widths, rather than allowing the number of rectangles to approach infinity, we take the longest subinterval and allow it to approach zero

This longest subinterval is called the

norm of the partition and denoted by

Regardless of the way we choose

and each value of

, all Riemann sums converge to a common value as all subintervals tend to zero 6Slide7

The Definite Integral as a Limit of Riemann Sums

DEFINITION

Let

be a function defined on a closed interval

. For any partition

of

, let the numbers be chosen arbitrarily in the subintervals .If there exists a number such thatno matter how and the ’s are chosen, then is integrable on and is the definite integral of

over

.

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Theorem 1: Existence of Definite Integrals

THEORM

All continuous functions are integrable. That is, if a function

is continuous on an interval

, then its definite integral over

exists.

(Note that, since the definition allows us to choose any partition, this theorem allows us to go back to using regular partitions for which all subintervals have the same length.) 8Slide9

Theorem 1: Existence of Definite Integrals

The Definite Integral of a Continuous Function on

Let

be continuous on

, and let

be partitioned into

subintervals of equal length, . Then the definite integral of over is given bywhere each is chosen arbitrarily in the th subinterval. Note that, since the norm of a partition is , then

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Terminology and Notation

Recall that we defined the derivative as

This notation is due to Leibniz, who switched from Greek letters to Roman letters to denote the derivative

He also created notation for integrals

The Greek capital sigma was chosen to represent sums, and an elongated Roman s represents the integral

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Terminology and Notation

We read this as “the integral from

a

to

b

of

f of x, dx” or as “the integral from a to b of f of x with respect to x”.Note that and are the endpoints of the interval and are written with the greater value on top and the lesser value on the bottom (we will see later than we can reverse this with an appropriate correction) 11Slide12

Terminology and Notation

The greater value is called the

upper limit of integration

; the lesser value is the

lower limit of integration

The symbol

is called the integral signThe function f is called the integrandThe notation dx denotes that x is the variable of integration 12Slide13

Terminology and Notation

Because the value of a definite integral depends only on the function, the letter we use as the variable of integration does not matter and is called the

dummy variable

Hence, for a given function

f

over an interval

 13Slide14

Terminology and Notation

Finally, a note about the difference between an indefinite integral (or antiderivative) and a definite integral:

Taking an indefinite integral results in

a family of functions that differ by a constant

Taking a definite integral results in

a real number

You will see how these are related to each other and to derivatives in the next section 14Slide15

Example 1

The interval

is partitioned into

subintervals of equal length,

. Let

denote the midpoint of the

th subinterval. Express the limitas an integral. 15Slide16

Example 1

Using

over

we get

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Area Under a Curve

DEFINITION

If

is nonnegative and integrable over a closed interval

, then the

area under the curve

from to is the integral of from to ,(Note that we can use integrals to calculate area, and we can use area to calculate integrals.) 17Slide18

Example 2

Evaluate the integral

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Example 2

The graph is the upper half of a circle (i.e., a semi-circle) with radius 2. Since we know that the area of semi-circle is

and that the area under the curve is

then

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Definite Integral and Area

The value of an integral over which a continuous function

has negative values is a negative number

Of course, this interval also covers an areaBut since we want area to be a positive value, if

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Definite Integral and Area

The value

of integrable functions that have both positive and negative values on

is

area above

-axis

area below -axis 21Slide22

Theorem 2: Constant Functions

THEOREM

If

, where

is a constant, over

, then

PROOFUsing a Riemann Sum with regular partitions, we have . The definite integral is then 

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Theorem 2: Constant Functions

THEOREM

PROOF

Note that

is a constant with respect to summation, so

We conclude that

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Theorem 2 Viewed Geometrically

It’s easy to show that theorem 2 holds by using geometry

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Example 3: A Train Problem

A train moves along a track at a constant velocity of 75 mph from 7:00 AM to 9:00 AM. Express its total distance traveled as an integral. Evaluate the integral using Theorem 2.

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Example 3: A Train Problem

By Theorem 2, with

,

, and

More importantly, note the units:

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Example 3: A Train Problem

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Example 3: A Train Problem

Let’s try to make some connections among the things you have learned

The integral is defined as the limit of a Riemann Sum when the number of rectangles approaches infinity

A Riemann Sum gives the area under a curve (or the negative value of that area if the curve lies under the x-axis)In the train problem, the integral is a distance, specifically, the distance that the train traveled from 7:00 AM to 9:00 AMBut distance traveled over a given period of time is given by the

position

function, not the velocity function

On the other hand, we know that the velocity function is obtained as derivative of the position functionWhat seems to be the relation between the definite integral ( and hence, the area under the curve) of a velocity function and the position of the train?28Slide29

Integrals on a Calculator

As we proceed through this chapter, and later on the AP exam, there will be instances where it is either very difficult or impossible to calculate a definite integral

You

must learn how to use your calculator to determine integralsOn a TI84, finding a definite integral is intuitive; just remember that this function is accessed through the MATH key29Slide30

Integrals on a Calculator

Directions for using the TI89 are below (from the TI89 manual, which can be downloaded from the internet)

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Example 4: Evaluating Definite Integrals on a Calculator

Use a calculator to evaluate the following integrals numerically.

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Example 4: Evaluating Definite Integrals on a Calculator

Evaluate the following integrals numerically.

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Example 4: Evaluating Definite Integrals on a Calculator

Integrals can also be evaluated from the Graph screen. This will not only give the value of the integral between the limits of integration, but will also shade the area.

Note that

is the basic form of the Gaussian curve (or the normal curve). It turns out that it is impossible to find an explicit value for this integral. If we want to know the area under the curve, we

must

approximate it using a Riemann Sum (and the calculator does this much more efficiently that any human ever could). You will encounter many more functions like this, so knowing how to use the calculator is essential.

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Discontinuous Integrable Functions

By Theorem 1, a function that is continuous over an interval is guaranteed to be integrable

That is, continuity implies

integrability (compare this with what we learned about derivatives: differentiability implies continuity)But it is sometimes the case that a function that has a discontinuity over some interval can still be integratedIn general, some functions with jump discontinuities or point discontinuities (but not unbounded discontinuities) are integrable

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Example 5: Integrating a Discontinuous Function

Find

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Example 5: Integrating a Discontinuous Function

We use the fact that the area (actually, the

net area)

under a curve over a given interval is equal to the definite integral evaluated over the interval.The given function is equal to 1 for

and to

for

; it is not defined at and this is where the discontinuity occurs. 36Slide37

Example 5: Integrating a Discontinuous Function

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Example 5: Integrating a Discontinuous Function

Since the net area (as determined by

and knowing that the area between

and

is negative) is

, then

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Definite Integral as an Accumulator Function

Example 1 from section 6.1 asked you to find the amount of snow that had fallen from 3 A.M. to 10:30 A.M.

This was solved by finding the area under the graph showing the rate of snowfall over this interval of time

If

represents the amount of snow that has fall by time

, then we can express this as an integral

where is the rate of snowfall at time  39Slide40

Definite Integral as an Accumulator Function

The definite integral makes it possible to build new functions called

accumulator functions

Note that this is a function of the value of

That is,

is a function of

, which is the upper limit of integration in the integralIn this case, we use a different variable of integration, , which is a dummy variable 40Slide41

Definition: Accumulator Function

DEFINITION:

If the function

is integrable over the closed interval

, then the definite integral of

from

defines a new function for , the accumulator function 41Slide42

Example 4: An Accumulator Function

Let

for

. Write an accumulator function

as a polynomial in

where

 42Slide43

Example 4: An Accumulator Function

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Example 4: An Accumulator Function

The area over the interval is a trapezoid with height

and bases

and

. The area of the trapezoid is the accumulation from 1 to

over the interval

. This area is 44Slide45

Exercise 6.2

Online

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