Fundamental Theorem of Calculus
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Transcript of Fundamental Theorem of Calculus
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?