Section 4.3 Riemann Sums and The Definite Integral

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Section 4.3 Riemann Sums and The Definite
Integral
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Area under a curve

• The other day we found the area under a curve by
  dividing that area up into equal intervals
  (rectangles of equal width).
• This section goes one step further and says we
  can divide the area into uneven intervals, and
  the same concept will still apply.
• It will also examine functions that are
  continuous, but are no longer entirely
  non-negative.

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Lets quickly review what we learned last time
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If f(x) is a nonnegative, continuous function on
the closed interval a, b, then the area of the
region under the graph of f(x) is given by
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Notice that through the use of rectangles of
equal width, we were able to estimate the area
under a curve. This curve was always
non-negative (i.e. it was always above the
x-axis) therefore, the values we got when we
multiplied f(xi) by ?x were always positive.
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The graph of a typical continuous function y
ƒ(x) over a, b Partition a, b into n
subintervals a lt x1 lt x2 ltxn lt b. Select any
number in each subinterval ck. Form the product
f(ck)?xk. Then take the sum of these products.
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• The sum of these rectangular areas is called the
  Riemann Sum of the partition of ?x.LRAM, MRAM,
  and RRAM are all examples of Riemann Sums.
• The width of the largest subinterval of a
  partition ? is the norm of the partition, written
  x.
• As the number of partitions, n, gets larger and
  larger, the norm gets smaller and smaller.
• As n??, x ?0

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Formal Definition from p. 266 of your textbook
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Finer partitions of a, b create more rectangles
with shorter bases.
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This limit of the Riemann sum is also known as
the definite integral of f(x) on a, b
This is read the integral from a to b of f of x
dx, or the integral from a to b of f of x with
respect to x.
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Notation for the definite integral
upper limit of integration
Integration Symbol(integral)
integrand
variable of integration(differential)
lower limit of integration
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Formal Definition from p. 267 of your textbook
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NOW Before we go any further, lets make one
quick yet important clarification regarding
INDEFINITE INTEGRALS (with which we have worked
before in finding antiderivatives) and DEFINITE
INTEGRALS (which we are about to study now)
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Today we will focus solely on the DEFINITE
integral. The connection between the two types
of integrals will probably not be clear now, but
be patient. We will explore and examine the
connection between the two next time. ? Woo-hoo!
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Evaluate the following definite integrals using
geometric area formulas.
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Top half only!
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THEOREM If f(x) is continuous and
non-negative on a, b, then the definite
integral represents the area of the region under
the curve and above the x-axis between the
vertical lines x a and x b.
a
b
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Rules for definite integrals
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The Integral of a Constant

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Rules for definite integralsPreservation of
Inequality Theorem
If f is integrable and non-negative on a, b then
If f and g are integrable and non-negative on
a, b and f(x) lt g(x) for every x in a,b, then
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Using rules for definite integrals
Example
Evaluate the using the following values
60 2(2) 64
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Using the TI 83/84 to check your answers

• Find the area under on 1,5
• Graph f(x)
• Press 2nd CALC 7
• Enter lower limit 1
• Press ENTER
• Enter upper limit 5
• Press ENTER.

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When functions are non-negative, the Riemann sums
represent the areas under the curves, above the
x-axis, over some interval a, b. When
functions are negative, however, the Riemann sums
represent the negative (or opposite) values of
those areas. In other words, Riemann sums DO
have direction and CAN take on negative values.
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To summarize that thought
f
A
a
b
A1
f
A3
area above area below
a
b
A2
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ASSIGNMENT
p. 272 273 (14-22even, 24-30even, 31,
34-40even, 45)