5'2b The Definite Integral

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5.2b The Definite Integral

• 4 Interpret definite integrals as AREAS
• The Midpoint Rule
• 5 Using the midpoint rule to compute ln2
• Properties of the definite integral
• 6 Use properties of integrals and formulas for
  sums of powers.
• 7 Illustrate properties of the integral.
• 8 Bound an integral.

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5.2 5 Midpoint Sum
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5.2 Algebraic Properties of the Integral
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5.2 Comparison Properties of the Integral
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5.3a Mean Value Theorem
THEOREM. If the function f is continuous on
a, b, then
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5.3a Mean Value Theorem
PROOF. By the Heine-Borel Theorem, f has a
minimum m and a maximum M on a, b. Then
for every x in a, b
But by the intermediate value theorem, f takes on
every value between m and M on a,b. That means
that