Fundamental Theorem of Calculus

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Fundamental Theorem of Calculus

• Finally!

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Objective

• To integrate using the Fundamental Thm of Calc

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Pandoras box
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Fundamental Thms

• The Fundamental Theorem of Arithmetic
• Any positive integer can be represented in
  exactly one way as a product of primes.
• The Fundamental Theorem of Algebra
• Every polynomial of degree n has exactly n
  zeroes.
• The Fundamental Theorem of Geometry
• No theorem wears this title, but perhaps the
  Pythagorean Theorem deserves it.

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Integrals area under the curve

• No problem if its a geometric shape (4.3)
• What if its not? How could we find the area
  under the curve?

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Rectangles
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An easier example.

• This is called Riemann Sums
• Using left-hand endpoints with 4 rectangles
• Area

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What if.

• We use right-hand endpoints and 4 rectangles?
• Area

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Whats a more accurate way to find area?
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How many rectangles is the best?
f(x) y- value or height and ?x (b-a)/n (n is
the number of rectangles)
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Riemann Sums and definite integrals
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Fundamental Theorem of Calculus

• If f is cont on a,b and F is an antiderivative
  of f on a,b then

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Example
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What about a C?
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Absolute values
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A different example

• Find the area of the region bounded by y2x2
  3x 2, x-axis, x 0, and x 2.
• Step 1 draw graph

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Ex cont

• Find the area of the region bounded by y2x2
  3x 2, x-axis, x 0, and x 2.
• Step 2 Write the integral and integrate

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Average Value of a function

• Average value

Find the average value of f(x) 3x2 2x on
1,4
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Pg 283, 31

• A company purchases a new machine for which the
  rate of depreciation is dV/dt 10,000(t-6) where
  0lt tlt 5 and V is the value of the machine after t
  years. What is the total loss of value of the
  machine over the first 3 years?

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