5'2 The Definite Integral

1
5.2 The Definite Integral
2
Review

• Evaluate

3
The Definite Integral

• If f is a continuous function defined for a x
  b.

• Divide the interval a, b into n subintervals
  of equal width of
• ?x (b a) / n.

? x2
? x1
? x3
? x4
a
b

• Let x1, x2, ., xn be random sample points
  from these subintervals so that xi lies anywhere
  in the subinterval xi-1, xi.

4
The Definite Integral
Then the definite integral of f from a to b is

? x2
? x1
? x3
? x4
a
b



Riemann Sum Bernhard Riemann (1826 1866)
The limit of a Riemann Sum as n ? 8 from x a
to x b.
5
The Definite Integral
A pattern cannot be developed if we use random
sample points, so we will use the left end, right
end, or midpoint of each subinterval. We drop the
asterisk in this case.
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Left, Right, Midpoint Rules

• Left
• Right
• Midpoint

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Examples
1. Use the Midpoint Rule with the given value of
n to approximate the integral. Round your answer
to four decimal places.
2. Write the limit of a Riemann Sum as a definite
integral on the given interval.
8
Examples
3. Use the form of the definition of the
definite integral to evaluate the integral.
4. Express the integral as a limit of a Riemann
Sum, but do not evaluate.
5. Evaluate the integrals by interpreting in
terms of area.
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Basic Properties of the Integral
Let a, b, and c be constants and f and g be
continuous functions on a, b.
3. Definite integrals can be positive or
negative.
4. Not all definite integrals can be
interpreted in terms of area, but definite
integrals can be used to determine area.
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Properties of the Integral
Let a, b, and c be constants and f and g be
continuous functions on a, b.
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Properties of the Integral
Let a, b, and c be constants and f and g be
continuous functions on a, b.
5. If f (x) 0 for a x b, then
6. If f (x) g (x) for a x b, then
7. If m f (x) M for a x b, then
12
Examples
6. Use the properties of integrals to verify the
inequality without evaluating the integrals.