AP CALCULUS AB

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AP CALCULUS AB

• Chapter 5
• The Definite Integral
• Section 5.2
• Definite Integrals

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What youll learn about

• Riemann Sums
• The Definite Integral
• Computing Definite Integrals on a Calculator
• Integrability
• and why
• The definite integral is the basis of integral
  calculus, just as the derivative is the basis of
  differential calculus.

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Sigma Notation
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Section5.2 Definite Integrals

• Definition of a Riemann Sum
• f is defined on the closed interval a, b, and
  is a partition of a, b given by
• where is the length of the ith
  subinterval.
• If ci is any point in the ith subinterval, then
  the sum
• is called a Riemann Sum of f for the partition
  .

a b
Partitions do not have to be of equal width If
the are of equal width, then the partition is
regular and
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The Definite Integral as a Limit of Riemann Sums

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The Existence of Definite Integrals

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The Definite Integral of a Continuous Function on
a,b

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The Definite Integral

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Section 5.2 Definite Integrals

• If f is defined on the closed interval a, b and
  the limit
• exists, then f is integrable on a, b and the
  limit is denoted by
• The limit is called the definite integral of f
  from a to b. The number a is the lower limit of
  integration, and the number b is the upper limit
  of integration.

The function is the integrand
x is the variable of integration
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Example Using the Notation

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Section 5.2 Definite Integrals

• Theorem If yf(x) is
• nonnegative and integrable
• over a closed interval a, b,
• then the area under the curve
• yf(x) from a to b is the
• integral of f from a to b,
• If f(x)lt 0, from a to b (curve is under the
  x-axis),
• then

a b
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Area Under a Curve (as a Definite Integral)

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Area

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The Integral of a Constant

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Section 5.2 Definite Integrals

• To find Total Area Numerically (on the
  calculator)
• To find the area between the graph of yf(x) and
  the x-axis over the interval
• a, b numerically, evaluate
• On the TI-89
• nInt (f(x), x, a, b)
• On the TI-83 or 84
• fnInt (f(x), x, a, b) Note use abs under
  MathNum for absolute value

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Example Using NINT (FnInt)

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Example Using NINT (FnInt)

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Discontinuous Functions
The Reimann Sum process guarantees that all
functions that are continuous are integrable.
However, discontinuous functions may or may not
be integrable. Bounded Functions These are
functions with a top and bottom, and a finite
number of discontinuities on an interval a,b.
In essence, a RAM is possible, so the integral
exists, even if it must be calculated in pieces.
A good example from the Finney book is f(x)
x/x.
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Discontinuous Functions
An example of a discontinuous function (badly
discontinuous), which is also known as a
non-compact function, is given also This
function is 1 when x is rational, zero when x is
irrational. On any interval, there are an
infinite number of rational and irrational values.