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?