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Fundamental Theorem of Calculus. Finally!. Objective…. To integrate using the Fundamental Thm of Calc. Pandora’s box…. Fundamental Thms. The Fundamental Theorem of Arithmetic: Any positive integer can be represented in exactly one way as a product of primes.

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Presentation Transcript
objective
Objective…
  • To integrate using the Fundamental Thm of Calc
fundamental thms
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.
integrals area under the curve
Integrals… area under the curve
  • No problem if it’s a geometric shape… (4.3)
  • What if it’s not? How could we find the area under the curve?
an easier example
An easier example….
  • This is called Riemann Sums
  • Using left-hand endpoints with 4 rectangles
  • Area =
what if
What if….
  • We use right-hand endpoints and 4 rectangles?
  • Area =
how many rectangles is the best
How many rectangles is the best?

f(x) = y- value or height and Δx = (b-a)/n (n is the number of rectangles)

fundamental theorem of calculus1
Fundamental Theorem of Calculus
  • If f is cont on [a,b] and F is an antiderivative of f on [a,b] then
a different example
A different example
  • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2.
  • Step 1… draw graph
ex cont
Ex cont…
  • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2.
  • Step 2: Write the integral and integrate
average value of a function
Average Value of a function
  • Average value =

Find the average value of f(x) = 3x^2 – 2x on [1,4]

pg 283 31
Pg 283, #31
  • A company purchases a new machine for which the rate of depreciation is dV/dt = 10,000(t-6) where 0< t< 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?