Integration The area under a curve

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When you see this symbol
Copy the notes and diagrams into your jotter.
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The area under a curve
Let us first consider the irregular shape shown
opposite.
How can we find the area A of this shape?
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The area under a curve
We can find an approximation by placing a grid of
squares over it.
By counting squares,
A gt 33 and A lt 60
i.e. 33 lt A lt 60
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The area under a curve
By taking a finer mesh of squares we could
obtain a better approximation for A.
We now study another way of approximating to A,
using rectangles, in which A can be found by a
limit process.
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The area under a curve
The diagram shows part of the curve y f(x) from
x a to x b.
A
We will find an expression for the area A bounded
by the curve, the x-axis, and the lines x a and
x b.
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The area under a curve
The interval a,b is divided into n sections of
equal width, ?x.
A
n rectangles are then drawn to approximate the
area A under the curve.
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The area under a curve
Dashed lines represent the height of each
rectangle.
The position of each line is given by an
x-coordinate, xn.
The first rectangle has height f(x1)
x1, x2 , x3, x4 , x5, x6
and breadth ?x1.
Thus the area of the first rectangle f(x1).?x1
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The area under a curve
An approximation for the area under the curve,
between x a to x b, can be found by summing
the areas of the rectangles.
A f(x1).?x1 f(x2).?x2 f(x3).?x3
f(x4).?x4 f(x5).?x5 f(x6).?x6
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The area under a curve
Using the Greek letter S (sigma) to denote the
sum of, we have
For any number n rectangles, we then have
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The area under a curve
In order to emphasise that the sum extends over
the interval a,b, we often write the sum as
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The area under a curve
By increasing the number n rectangles, we
decrease their breadth ?x.
As ?x gets increasingly smaller we say it tends
to zero, i.e. ?x ? 0.
Remember, we met limits before with
Differentiation
So we define
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The area under a curve
was simplified into the form that we are familiar
with today
This reads
the area A is equal to the integral of f(x) from
a to b.
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The area under a curve
We have derived a method for finding the area
under a curve and a formal notation
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The area under a curve
Let us remind ourselves of where we started.
Can we apply this method to calculate the area
under a curve?
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The area under a curve
Consider a strip under the curve h wide.
The inner rectangle has area h ? f(x).
The outer rectangle has area h ? f(xh).
The actual area is given by A(xh) A(x).
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The area under a curve
Comparing areas,
h ? f(x) ? A(xh) A(x) ? h ? f(xh)
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The area under a curve
h ? f(x) ? A(xh) A(x) ? h ? f(xh)
So f(x) A(x), by the definition of a derived
function
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The area under a curve
In conclusion,
the area A bounded by the x-axis, the lines x a
and x b and the curve y f(x) is denoted by,