Area and the Definite Integral

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Area and the Definite Integral
Tidewater Community College Mr. Joyner, Dr. Julia
Arnold and Ms. Shirley Brown using Tans 5th
edition Applied Calculus for the managerial ,
life, and social sciences text
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Area and the Definite Integral
Suppose a states annual rate of oil consumption
each year over a 4-year period is a constant
function, such as f(t) 2, where t is measured
in years and f(t) is measured in millions of
barrels. We would graph such a function as
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Area and the Definite Integral
How could you use the graph to answer the
question, How much oil was used during the four
years?
Answer 2(4-0) 8 million barrels
It seems reasonable that the answer is 8 million
barrels and this is the same as the area of the
rectanglethe area under the curve. (Yes, even
a straight line is considered a curve in
mathematics!)
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Area and the Definite Integral
How would you find the area above the x-axis and
under the curve f(x) x from x 0 to x 4?
Does it seem reasonable to find the area of the
triangle the area under the curve.
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Area and the Definite Integral
Imagine now that our function, f(x) x,
represents the oil consumption of the same state
over the same four years.
What was the oil consumption over the four years?
Lets explore.
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Area and the Definite Integral
For the first year, the state started off
consuming no oil (zero barrels per year), and
ended the year consuming oil at the rate of 1
million barrels per year.
Since the rate of increased consumption is
linear, we can use the average rate of change
over the year to represent the entire first year.
The average rate of change over the first year is
(1 0)/2 0.5 million barrels per year.
For one year, (the first year) the amount of oil
consumed is 1(0.5) 0.5 million barrels
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Area and the Definite Integral
For the first two years, the state started off
consuming no oil and ended the two-year year
period consuming oil at the rate of 2 million
barrels per year.
Again, since the rate of increased consumption is
linear, we can use the average rate of change
over the 2 years to represent the entire two-year
period.
The average rate of change over the 2-year period
is (2 0)/2 1 million barrels per year.
For two years. the amount of oil consumed is
2(1) 2 million barrels.
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Area and the Definite Integral
In a similar fashion we can reason that the total
oil consumption is as follows
Lets compare these values to
the areas of the right triangles under the curve
f(x).
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Area and the Definite Integral
Triangle areas
We get the same values!
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Area and the Definite Integral
How would we find the area under the following
curve and above the x-axis?
The work is a little more complex, but the
approach is the same
area under the curve!
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Area and the Definite Integral
We could use rectangles to approximate the area.
First, divide the horizontal base with a
partition. We will start with a partition that
has two equal subintervals 0,1 and 1,2
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Area and the Definite Integral
In this slide, the subinterval rectangles have
been drawn using the right-hand endpoints of each
subinterval. We could have used the left
endpoints, the midpoints of the intervals, or any
point in each subinterval.
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Area and the Definite Integral
The area of the first rectangle is 2 and is
greater than the area under the curve.
The area of the second rectangle is 5 and is also
greater than the area under the curve.
5
2
1
1
0
2
1
The total area of the rectangles is 2 5 7
which is greater than the desired area under the
curve.
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Area and the Definite Integral
In this slide, lets draw the rectangles using
the left-hand endpoints.
12 1 2
02 1 1
0
2
1
1unit
1unit
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Area and the Definite Integral
Now, lets calculate the areas of these new
rectangles.
The area of the first rectangle is 1 x 1 1
and is less than the area under the curve.
The area of the second rectangle is 1 x 2 2
and is also less than the area under the curve.
The total area of the rectangles using the
left-hand endpoints is 1 2 3 which is less
than the desired area under the curve.
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Area and the Definite Integral
Now we know that the area under the curve must be
between 3 and 7. Can we get a more accurate
answer? Yes! And to do so, we need to construct
more subintervals using a finer partition.
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Area Under A Curve
Area and the Definite Integral

• The area under a curve on an interval can be
  approximated by summing the areas of individual
  rectangles on the interval.

• If the rectangles are inscribed under the curve,
  the approximated area will be less than the
  desired area.
• (In our example, this area was 3.)

• If the rectangles are circumscribed above the
  curve, the area will be greater than the desired
  area.
• (In our example, this area was 7.)

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Area and the Definite Integral
Area Under A Curve

• By using a finer partition (one with more
  subintervals), we make each rectangle narrower
  (increasing their number), and we get an area
  value that is closer to the true area under the
  curve.
• By taking the limit as n (the number of
  rectangles) approaches infinity, the actual area
  is approached.

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Area and the Definite Integral
Area Under A Curve
Definition A sum such as the one below is
called a Riemann sum
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Area and the Definite Integral
Area Under A Curve
The area under a curve can be approached by
taking an infinite Riemann sum.
Let f(x) be a nonnegative, continuous function on
the closed interval a, b. Then, the area of
the region under the graph of f(x) is given
by where x1, x2, x3, xn are arbitrary
points in the n subintervals of a,b of equal
width .
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Area and the Definite Integral
On the following six slides, you will see the
number of rectangles increasing for the graph of
y x3 on the interval -1, 2 . Observe the way
that the area of rectangles more closely
approximates the area under the curve as the
number of rectangles increases.
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Area and the Definite Integral
As you can see, the sum of the areas of the
rectangles increases as you increase the number
of rectangles.
When there are 10 rectangles, Area 2.4675
20 rectangles, Area 3.091875 40
rectangles, Area 3.416719 100
rectangles, Area 3.615675 1000
rectangles, Area 3.736507 5000
rectangles, Area 3.7473
As the number of rectangles increases, they begin
to better fit the curve giving a closer and
closer approximation of the area (3.75) under the
curve and above the x-axis.
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Area and the Definite Integral
We are almost done with this lesson in that we
can now turn our attention to the definition and
discussion of the definite integral for all
functions, not just nonnegative functions.
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Area and the Definite Integral
Definition The Definite Integral
Let f(x) be defined on a,b. If exists for
all choices of representative points x1, x2, x3,
xn in the n subintervals of a,b of equal
width , then this limit is
called the definite integral of f(x) from a to b
and is denoted by .
The number a is the lower limit of integration,
and the number b is the upper limit of
integration.
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Area and the Definite Integral
Definition The Definite Integral
Thus
As long as f(x) is a continuous function on a
closed interval, it has a definite integral on
that interval. f(x) is said to be integrable
when its integral exists.
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Area and the Definite Integral
The Definite Integral
Please make this important distinction between
the indefinite integral of a function and the
definite integral of a function
The indefinite integral of a function is another
function. example
The definite integral of a function is a number.
example
You will learn how to calculate this number in
the next section of your textbook.
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Area and the Definite Integral
The Definite Integral
Lets take a quick look at the geometric
interpretation of the definite integral, and
well be done.
If f(x) is a nonnegative, continuous function on
a, b, then is equal to the
area of the region under the graph of f(x) on a,
b.
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Area and the Definite Integral
The Definite Integral
If f(x) is a nonnegative, continuous function on
a, b, then is equal
to the area of the region under the graph of f(x)
on a, b.
What happens if f(x) is not always nonnegative?
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Area and the Definite Integral
The Definite Integral
If f(x) is simply a continuous function on a,
b, then is equal to the
area of the region below the graph of f(x) and
above the x-axis minus the area of the region
above the graph of f(x) and below the x-axis on
a, b.
Area of region 1 Area of region 2 Area of
region 3
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Area and the Definite Integral
The Definite Integral
Lets look at a problem similar to 3 of the
section 6.3 Exercises (page 475) in the textbook.
We shall use right endpoints.
Let f(x) 3x. Part a Sketch the region R
under the graph of f(x) on the
interval 0,2 and find its exact area using
geometry.
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Area and the Definite Integral
The Definite Integral
Solution - Part a Since the area under the curve
from 0,2 forms a right triangle, we can
calculate the exact area using the geometry
formula A ½ bh b 2 and h 6, so ½ (2)(6) 6
6
f(x) 3x
4
2
Area 6
0
2
1
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Area and the Definite Integral
The Definite Integral
Let f(x) 3x. Part b Use a Riemann sum with
four subintervals (n 4) of equal length to
approximate the area of region R. Choose the
representative points to be the right endpoints
of the subintervals.
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Area and the Definite Integral
Solution Part b Divide the length of the
interval (2) by the number of subintervals (4).
Each subinterval has a length equal to ½ unit
. This produces the 4
subintervals 0, 1/2, 1/2, 1,
1, 3/2, and 3/2, 2 .
The Riemann Sum is A f(1/2) f(1) f(3/2)
f(2)(1/2)
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Area and the Definite Integral
The Definite Integral
Let f(x) 3x. Part c Repeat part (b) with
eight subintervals (n 8) of equal length.
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Area and the Definite Integral
Solution Part c Divide the length of the
interval (2) by the number of subintervals (8).
Each subinterval has a length equal to 1/4 unit
. This produces the 8
subintervals
0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1 . 1,
5/4, 5/4, 3/2, 3/2, 7/4, and 7/4, 2
The Riemann Sum is A f(1/4) f(1/2) f(3/4)
f(1) f(5/4) f(3/2) f(7/4) f(2)(1/4)
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Area and the Definite Integral
The Definite Integral
Let f(x) 3x. Part d Compare the
approximations obtained in parts (b) and (c)
with the exact area found in part (a). Do
the approximations improve with larger n ?
Solution Part d Compare the values. What is
you conclusion? Hopefully, you see that the
approximation does improve with a finer partition
(and thus, larger values of n).