The Definite Integral

Day 2 – Trapezoidal Approximations of

Area Between Curve and x-axis.

Area Under Curves

How could we make these approximations more accurate?

Today’s Learning Outcomes

Approximate area between curve and x-axis

using trapezoids when:

-Given a graph of a curve

Connect trapezoidal approximations

to rectangular approximation methods

Relate function behavior to trapezoidal approximations

GEOMETRY Review

What is a trapezoid?

What does a trapezoid look like?

What is the formula for the finding the area of trapezoid?

Quadrilateral with one pair of parallel lines.

These parallel sides are called the bases.

1

2h b1 +b2( )

Let’s examine this picture . . .

The curve is the blue wave.

There are four trapezoids drawn.

Can you identify the bases of each trapezoid?

Today . . .

We will approximate the area under a curve using trapezoids.

Trapezoidal Rule (packet 3-4)

1. Using the function from [0,3] to

the right, finish drawing in trapezoids by

connecting the endpoints of the

partitions. Assume there are three

partitions.

Together lets label the bases of the

trapezoids (which are all sideways):

Trapezoid #1

Trapezoid #2

Trapezoid #3

Note: This first trapezoid is a

special case!! It is a triangle

which is a trapezoid with a

base =0.

y = x2

Trapezoidal Rule (packet 3-4)

2. Looking at these, do you think adding up

the trapezoids gives a more accurate or

less accurate area estimate than the

rectangles we did yesterday? WHY?

3. Write the formula for area of a trapezoid.

We did this a minute ago!!

Trapezoidal Rule

4. Use this formula to estimate the area

of the three trapezoids in the drawing to

the right. Add them up to get an estimate

for the area under this curve.

Trapezoidal Rule—YOU TRY!

5. Use the same concept to estimate the area under the curve

from [0,9]. USE PARTITIONS OF LENGTH 3.

y x

Trapezoidal Rule

7. Look back at today’s Warmup where we used LRAM

and RRAM to estimate these same areas. What

relationship does the Trapezoidal Area have with the two

of these? Why is that?

LRAM= 12.545

RRAM= 21.545

Trapezoidal (from previous slide) = 17.045

Other items--NOTES

Trapezoidal estimate = LRAM + RRAM

2

When showing work, show the formula with the actual

numbers substituted in.

DON’T make the common mistake!!!

MRAM 2

LRAM RRAM

Today’s Learning Outcomes

Checking your understanding

Approximate area between curve and x-axis

using trapezoids when:

-Given a graph of a curve

Connecting trapezoidal approximations

to rectangular approximation methods

Relating function behavior to trapezoidal approximations

Thumbs up?

More Learning Outcomes

Approximate area between curve and x-axis

using trapezoids when:

-Given a table of data

Apply trapezoidal rule for approximating area

when given a uniform width

Brainstorm with your partner

• Can we come up with a rule or formula if we have uniform widths?

Trapezoidal Rule 6. Look back at your #4 and #5. Do you notice that in adding

the areas up, every partition’s height is used twice except

the first and the last. Try to write a general equation for

the area under a curve using trapezoids. Use b1 for

base 1, b2 for base 2, etc.

NOTES: Trapezoidal Rule Summary

A =1

2h b0 + 2b1 + 2b2 + ... + 2bn-1 + bn( )

h = width of partition

b = value of function at the specific x - value

Trapezoidal Rule--Example

8. Coal gas is produced at gasworks. Pollutants in the gas are

removed by scrubbers, which become less and less efficient as

time goes on. The following measurements, made at the start

of each month, show the rate at which pollutants are escaping

(in tons/month) in the gas:

Use trapezoidal rule to estimate the quantity of pollutants that

escaped during the first three months. During all six months.

Trapezoidal Rule—FINISH #9 and #10 TONIGHT 9. Fred is driving down Rock Quarry Road when his radar system warns him of

an obstacle 400 feet ahead. He immediately applies the breaks, starts to slow

down, and spots a skunk in the road directly in front of him. The “black box”

data in the Porsche records the car’s speed every two seconds, producing the

following data. The speed decreases throughout the 10 seconds it takes to

stop, although not necessarily at a uniform rate.

a) What is your estimate, using trapezoidal rule, of

the total distance that the student traveled

before coming to rest?

b) Based on this answer, do you think he hit the

skunk? (Assume the skunk did not run out the

road fast enough!)

Approximate area between curve and x-axis

using trapezoids when:

-Given a table of data

Apply trapezoidal rule for approximating area

when given a uniform width

Learning Outcomes

Checking your understanding

Thumbs up?

MORE Learning Outcomes

Identify the integral symbol and what it

represents

Evaluate integrals using fnINT

Notes

EXACT area between a curve, y=f(x), and x-axis from a to b is

the

INTEGRAL of f from a to b:

( )

b

a

Area f x dx

x4

2

6

ò dx Means find the area between

the x-axis and y = x4 over the

interval [2,6]

Example:

Important

• If the value of an integral is negative the area would be below the x-axis.

• In this case, the actual area would be the absolute value of each integral.

• We will use the calculator to explore this idea.

fnInt on the Calculator

fnInt(y1, x,a, b)

Function to integrate Lower bound

Upper bound

Integration variable

MATH #9

OR on NEW Operating System,

HOT button at the top of the calculator key pad

ALPHA F2

fnInt on the Calculator

3

6

0Evaluate

f x x

f x dx

fnInt(y1, x, 0, 6)

Function to integrate Lower bound

Upper bound

Integration variable

fnInt on the Calculator 3

0

2Evaluate

f x x

f x dx

fnInt(y1, x, -2, 0)

Function to integrate Lower bound

Upper bound

Integration variable

0 6

3 3

2 0

_____ ______x dx x dx

BUT the area between f(x)=x3 and the x-axis is:

Identify the integral symbol and what it

represents

Evaluate integrals using fnINT

MORE Learning Outcomes

How are you doing?

Exit ticket

• If a curve is concave down, then a trapezoidal approximation would yield a _____________ estimation.

• Trapezoidal Rule states that A=____________

• Can we have a negative actual area? Why/Why not?