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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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## Fundamental Theorem of Calculus

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**Fundamental Theorem of Calculus**Finally!**Objective…**• To integrate using the Fundamental Thm of Calc**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**• 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….**• This is called Riemann Sums • Using left-hand endpoints with 4 rectangles • Area =**What if….**• We use right-hand endpoints and 4 rectangles? • Area =**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 Calculus**• If f is cont on [a,b] and F is an antiderivative of f on [a,b] then**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…**• 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 = Find the average value of f(x) = 3x^2 – 2x on [1,4]**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?

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