Calculus 5.2

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5.3 Definite Integrals
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Find the area under the curve from x 1 to x 2.
Example
The best we can do as of now is approximate with
rectangles
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Find the area under the curve from x 1 to x 2.
Example
The best we can do as of now is approximate with
rectangles
Lets try 4 rectangles
? 1.97
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Remember from last time
So what do all of these symbols mean?
upper limit of integration
Integration Symbol
integrand
variable of integration
lower limit of integration
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Rectangle Height
Summation Symbol
Rectangle Base
Dont forget that we are still finding the area
under the curve. Now, how to evaluate the
Definite Integral
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Let F(x) be a function such that F'(x) f(x)
In other words, f is the derivative of F and F is
the anti-derivative of f.
From this we can also infer that
And therefore that

This will be useful
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Now lets get the notation straight
?x
and therefore
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Now lets get the notation straight
And if we remove the limit (making it an
approximation)
?x
and multiply both sides by ?x, we get
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k 1
k 2
Now lets write out each of the rectangles
k 3
? ? ?
Since we are basing this on the left hand
method, we stop at k 1
? ? ?
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Now when we we add up all of the rectangles, what
happens on the right side of the approximation?
? ? ?
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The Definite Integral
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Find the area under the curve from x 1 to x 2.
Example
Area under the curve from x1 to x2.
There can be a slight difference between finding
the total area under the curve and finding the
integral as we shall see next
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Example
Find the area between the x-axis and the
curve from to .
pos.
neg.
AHA! But here we need to remember that area
under the x-axis will be negative. So lets be
careful what we are looking for.
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Example
Find the definite integral of from to
.
pos.
neg.
1 0
1
and notice that the lower area is 2 and the
upper area is 1. so the integral is the sum of
the two numbers not the sum of the actual areas.
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Page 269 gives rules for working with integrals,
the most important of which are
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4.
Integrals can be added and subtracted.