The Definite Integral

Day 1 – Areas Between Curves and x-axis

a.k.a—“Area Under Curves”

The Foundations to Calculus--LIMITS

•Remember the role that limits had in defining derivatives:

•Two limit definitions of a derivative

•Showing continuity to determine if a derivative even exists

The First Branch of Calculus:

Derivatives

•Where we have been—Derivatives

•Instantaneous rates of change lead to the exploration of

tangent lines and then to the derivative

•Describing how functions change at any given instant

The Second Branch of Calculus:

Integral Calculus •Where we are headed—Integration

•Describing how instantaneous changes accumulate over

an interval

•Investigation of area under curves

•LIMITS will again have a role in this branch of calculus

•Area BETWEEN curves

•Volume of Solids

Connecting the Branches:

•Mathematicians Newton and Leibniz knew the two concepts

of slopes of tangent lines and areas under curves were some

how connected

•The Fundamental Theorem of Calculus (Two Parts)

The Definite Integral

Day 1 – Areas Between Curves and x-axis

a.k.a—“Area Under Curves”

NOTES: Background Problem

• A car is traveling at a constant rate of 55 miles per hour from 2pm to 5pm. How far did the car travel?

▫ Numerical Solution—

▫ Graphical Solution—

• But of course a car does not usually travel at a constant

rate . . . .

BEFORE we consider that

possibility . . .

• We will introduce some new Calculus concepts

▫ LRAM--

▫ RRAM--

▫ MRAM--

Rectangular Approximation Method (RAM) to

estimate area under curve

Example Problem: Use LRAM with partition widths of 1

to estimate the area of the region enclosed between the

graph of and the x-axis on the interval [0,3].

NOTICE—This is an underestimate.

What if the function had been a

decreasing function?

2( )f x x

Is this an overestimate or an underestimate?

What if the function had been a decreasing

function?

BEWARE—MRAM uses the average of the

two x-values NOT the two y-values!

MRAM gives the best estimate regardless of

whether the function is increasing or decreasing.

CAUTION!!!!

MRAM ≠ (LRAM + RRAM)/2

Complete the following table:

Increasing

Function

Decreasing

Function

LRAM

RRAM

Let’s see if you understand directions . . .

Find the upper and lower approximations of the area under

on the interval [1,2] using 5 partitions.

( ) 2f x x

Find the upper and lower approximations of the area under

on the interval [0,2] using 8 partitions.

1( )f x

x

How do we get the best underestimate of the area

between a and b?

a b

Answer: Divide into 2 intervals (increasing/decreasing) and

pick LRAM or RRAM for each part that yields an

underestimate for that half.

A car is traveling so that its speed is never decreasing during

a 10-second interval. The speed at various points in time is

listed in the table below.

Example Problem: Packet p. 1

Time (seconds

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

“never decreasing”—

Sketch scatterplot:

A car is traveling so that its speed is never decreasing during

a 10-second interval. The speed at various points in time is

listed in the table below.

Example Problem: Packet p. 1

1. What is the best lower estimate

for the distance the car traveled

in the first 2 seconds?

“lower estimate”--_________ since increasing function

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

A car is traveling so that its speed is never decreasing during

a 10-second interval. The speed at various points in time is

listed in the table below.

Example Problem: Packet p. 1

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

2. What is the best upper estimate

for the distance the car traveled in

the first 2 seconds?

You Try: 3. What is the best lower estimate for the total distance

traveled during the first 4 seconds? (Assume 2 second

intervals since data is in 2 second intervals)

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 4. What is the best upper estimate for the total distance

traveled during the first 4 seconds?

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 5. Continuing this process, what is the best lower estimate

for the total distance traveled in the first 10 seconds?

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

You Try: 6. What is the best upper estimate for the total distance

traveled in the first 10 seconds?

Time (sec)

0 2 4 6 8 10

Speed (ft/sec)

30 36 40 48 54 60

Riemann Sums

These estimates are sums of products and are known as

Riemann Sums.

7. If you choose the lower estimate as your approximation of

how far the car traveled, what is the maximum amount your

approximation could differ from the actual distance?

SUMMARY

The approximations we calculated above are known as

______________________.

Each rectangle you create in this method is called a

___________________.

The accuracy of the approximation can be improved by

____________________________________.

Area Under Curves

How could we make these approximations more accurate?