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?