Area under a Curve
Formulas for areas of plane regions with straight-line boundaries (squares, rectangles, triangles, parallelograms,
trapezoids, etc…) have been well known in many civilizations.
But, regions with curvilinear boundaries caused problems for early mathematicians ( take a circle as the simplest
case).
Question: How did the early mathematicians find the area of a circle?
Basically, they used knowledge of the area of objects already known. We can do this by inscribing a regular polygon
INSIDE the circle, and as the number of sides increases the better approximation of the area we will have.
N=3
N=4
N=6
N=7
N=5
N=8
*** Key to finding the area is to take an N-sided polygon and subdivide it into N-congruent isosceles triangles.
Since we have N-congruent triangles of radius, r, the angle at the apex of one triangle is:
One Blown Up Triangle of radius r
r
r
One triangle of radius r has:
Base =
Height =
The AREA of one triangle can be written as:
Thus, the AREA of N triangles can be written as:
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Area under a continuous function f(x)
Our goal is to find the area under the graph of a function, f(x), over an interval where
and non-negative.
where f is continuous
Idea:
1. Divide the interval from a to b into
2. Over each
construct a
whose
height is the value of
at any point in the
3. The union (better yet, SUM) of these
forms a region, S,
whose
is an
of the region.
4. Repeat the process using more and more
Note: each sub-interval has width,
y
y = f(x)
a
b
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x
Example using rectangles to find area under a curve
Consider the function,
, on the interval where
(a) Break the region into 4 sub-intervals:
Approximate the area using 4 rectangles where the height of each rectangle is based on the Left endpoint of
each rectangle.
Approximate the area using 4 rectangles where the height of each rectangle is based on the Right endpoint of
each rectangle.
Approximate the area using 4 rectangles where the height of each rectangle is based on the Midpoint of each
rectangle.
(b) Break the region into 8 sub-intervals:
Approximate the area using 4 rectangles where the height of each rectangle is based on the Left endpoint of
each rectangle.
Approximate the area using 4 rectangles where the height of each rectangle is based on the Right endpoint of
each rectangle.
Approximate the area using 4 rectangles where the height of each rectangle is based on the Midpoint of each
rectangle.
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Summation
In estimating with finite sums, we want to take many terms, it is best to introduce a more convenient
notation for sums with larger terms.
SIGMA NOTATION allows us to write the SUM with many terms in a very compact form.
Use the Greek letter (capital Sigma) , ∑ , which represents SUM.
The index of the sum, k, tells us where the sum begins (number below)
and ends (number above). Note any letter can be used to denote the
index, but the letters, i , j, k are used most.
For example:
Important Identities: (these can be proved using the principle of mathematical induction, a proof
technique that is most often taught in a Discrete Mathematics course.)
1) Sum (add) the constant D a total of n times:
2) Sum (add) consecutive integer values:
3) Sum (add) consecutive integer values squared:
4) Sum (add) consecutive integer values cubed:
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Area under a continuous function f(x)
Our goal is to find the area under the graph of a function, f(x), over an interval where
and non-negative.
where f is continuous
Note: each sub-interval has width,
y
y = f(x)
a
b
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x
Example for Area under a continuous function
Consider the curve given by,
on the interval [0, 1].
(a) Find the Area under this curve on this interval using 5 equal sub-intervals
based on the Right-Hand Summation.
(b) Set-up the Area under this curve on this interval using n equal sub-intervals
based on the Right-Hand Summation.
(a) Take the limit as n approaches infinity. (we will have many thin rectangles)
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Example for Area under a continuous function
Consider the curve given by,
on the interval [-1, 1].
(a) Set-up the Area under this curve on this interval using n equal sub-intervals
based on the Right-Hand Summation.
(b) Take the limit as n approaches infinity. (we will have many thin rectangles)
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Example for Area under a continuous function
Consider the curve given by,
on the interval [-1, 2].
(a) Set-up the Area under this curve on this interval using n equal sub-intervals
based on the Right-Hand Summation.
(b) Take the limit as n approaches infinity. (we will have many thin rectangles)
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