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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. - PowerPoint PPT Presentation
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Fundamental Fundamental Theorem of Theorem of
CalculusCalculusFinally!Finally!
Objective…Objective…• To integrate using the Fundamental
Thm of Calc
Pandora’s box…Pandora’s box…
Fundamental ThmsFundamental 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 Integrals… area under the curvecurve
• No problem if it’s a geometric shape… (4.3)
• What if it’s not? How could we find the area under the curve?
Rectangles…Rectangles…
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 =
What’s a more accurate way to What’s a more accurate way to find area?find area?
How many rectangles is the How many rectangles is the best?best?
f(x) = y- value or height and Δx = (b-a)/n (n is the number of rectangles)
Riemann Sums and definite Riemann Sums and definite integralsintegrals
b
a
n
ii
ndxxfxxf )()(lim
1
Fundamental Theorem of Fundamental Theorem of CalculusCalculus
• If f is cont on [a,b] and F is an antiderivative of f on [a,b] then
)()()()( ] aFbFxFdxxfb
a
b
a
ExampleExample
2
1
2 )3( dxx
4
1
3 dxx
4
0
2sec
xdx
What about a + C?What about a + C?
b
adxxf )(
Absolute values…Absolute values…
2
012 dxx
A different exampleA 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 functionAverage Value of a function• Average value =
b
adxxf
ab)(
1
Find the average value of f(x) = 3x^2 – 2x on [1,4]
Pg 283, #31Pg 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?
