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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?
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
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
Identify the integral symbol and what it
represents
Evaluate integrals using fnINT
MORE Learning Outcomes
How are you doing?