Sigma Notation What does the following notation mean means the sum of the numbers from the lower number to the top number Area under curves In 51 we found that we can approximate areas using rectangles ID: 258856
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Slide1
5.2 Definite IntegralsSlide2
Sigma Notation
What does the following notation mean?
means
the sum of the numbers from the lower number to the top number.Slide3
Area under curves
In 5.1, we found that we can approximate areas using rectangles.
How do we get a more accurate approximation for area?
Add more rectangles that have smaller widths.
Adding all of these rectangles together gives is called a
Riemann Sum
.Slide4
Definition of a Riemann Sum
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 < x
3 < … < 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
Δ
x
i
be the width of the
i
th
interval [
x
i-1
, x
i
] and ci be any point in the
ith interval.Then, the Riemann sum of f for the partition is Slide5
Definition of a Definite Integral
As you add more and more rectangles under a curve, the widths, or partitions, of each rectangle become smaller and smaller.
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 Δ
x = (b – a)/n, then the definite integral of
f over [a, b] is given bySlide6
Existence of Definite Integrals
All continuous functions are
integrable.
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.Slide7
Leibniz’s is a Genius!!!! (thank goodness)
Leibniz introduced a notation for the definite integral that is much friendlier.
Riemann’s notation
Leibniz’s notationSlide8
Integration
Symbol
lower limit of integration
upper limit of integration
integrand
variable of
integrationSlide9
Areas under Curves
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.
If
f is nonpositive, then Area = Slide10
Areas under Curves
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, Slide11
Your Best Friend…
fnInt
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)Syntax:
(old operating system): fnInt(f(x), x
, a, b)(new operating system): Slide12
fnInt
Use
fnInt to find the following integrals: