Transcript 5.2 Definite Integrals

5.2 Definite Integrals
Sigma Notation
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What does the following notation mean?
5
n
 1 2  3  4  5
n 1

3
means the sum of the numbers from the
lower number to the top number.
 (2n  1)  (2(1) 1)  (2(0) 1)  (2(1) 1)  (2(2) 1)  (2(3) 1)
n  1
Area under curves
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In 5.1, we found that we can approximate areas using
rectangles.
How do we get a more accurate approximation for area?
3
Adding all of these
rectangles together
gives is called a
Riemann Sum.
2
1
0

1
2
3
4
Add more rectangles that have smaller widths.
Definition of a Riemann Sum
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Now imagine any function f that is defined on an interval
[a, b].
Let the notation xi’s points be on [a, b] such that x0 = a,
xn = b, and a < x1 < x2 < x3 < … < xn-1 < b.
The points a, x1, x2, x3, … , xn-1, xn, xn+1, b form a partion of
f notated as Δ on [a, b].
Let Δxi be the width of the ith interval [xi-1, xi] and ci be
any point in the ith interval.
Then, the Riemann sum of f for the partition is
n
 f (c )x
i 1
i
i
Definition of a Definite Integral
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As you add more and more rectangles under a curve, the
widths, or partitions, of each rectangle become smaller
and smaller.
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SOUND FAMILIAR????
This causes the area to be more and more accurate…so
accurate that it gives the actual area.
If f is continuous on [a, b] and [a, b] is partitioned into n
subintervals of equation length Δx = (b – a)/n, then the
definite integral of f over [a, b] is given by
n
lim
n 
 f (c )x
k 1
k
Existence of Definite Integrals
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All continuous functions are integrable.
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That is, if a function f is continuous on an interval [a, b], then its
definite integral over [a, b] exists.
These integrals calculate the area under a curve.
Leibniz’s is a Genius!!!! (thank goodness)
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Leibniz introduced a notation for the definite integral that
is much friendlier.
Riemann’s notation
n
lim
n 
 f (c )x
k 1
k
Leibniz’s notation
b
 f ( x)dx
a
upper limit of integration
Integration
Symbol

b
a
f  x  dx
integrand
variable of integration
lower limit of integration
Areas under Curves
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If an integrable function f(x) is nonpositive, the nonzero
terms in the Riemann sums for f over an interval [a, b]
are negatives of rectangle areas.
The integral of f from a to b is therefore the negative area
of the region between the graph and the x-axis.
b
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If f is nonpositive, then Area = 
 f ( x)dx
a
Areas under Curves
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b
If an integrable function f(x) has both positive and negative
values on an interval [a, b], then the Riemann sums for f
on [a, b] add areas that lie above the x-axis to the
negatives of areas that lie below the x-axis.
The resulting cancellations mean that the limiting value
(integral) is a number whose magnitude is less than the
total area between the curve and the x-axis.
Therefore, for any integrable function,
 f ( x)dx  (area above x - axis)  (area below x - axis)
a
Your Best Friend…fnInt
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We will eventually learn how to evaluate some definite
integrals by hand. However, not all definite integrals can
be evaluated by hand (at least, not without the help of
some geniuses who have done them first).
Your calculator has a function in it that will evaluate a
definite integral, which finds the area under a curve.
fnInt (MATH, #9)
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Syntax:
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(old operating system): fnInt(f(x), x, a, b)
(new operating system): b
 f ( x)dX
a
fnInt
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Use fnInt to find the following integrals:
2
 x sin xdx
1
1
4
0 1  x 2 dx
5
e
0
 x2
dx