# Definite Integral by dffhrtcv3

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```									Finite Sums

Where we have been: rates of change, slopes of tangent lines, derivatives
-differential calculus

How can all those rates of change accumulate
- areas under curves      -integral calculus

Newton and Leibniz proved connection between the two
The Fundamental THM of Calculus
-one of the most important insights ever!!

Estimating with Finite Sums
-remember problems from physics
traveling constant velocity at a given time interval
(distance = rate x time)
-area under the curve was the distance

This should work with varying velocity as well
-just break up the velocity into small time intervals
-then get a bunch of small rectangles - you can add up

Ex. A particle starts at x=0 and moves along the x-axis with velocity

For time greater than zero. Where is the
particle at 3 seconds?

Break into rectangles
-more used - more accurate

3 ways to determine top of rectangle
RAM - Rectangular Approximation Method
MRAM - Midpoint RAM
LRAM - Left hand RAM
RRAM - Right hand RAM

Sub Interval 0,1            1,2   2,3
Midpoint

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Midpoint
Height
Area

Sub          0,1        1,2        2,3
Interval
Left
Height
Area

Sub          0,1           1,2    2,3
Interval
Right
Height
Area

Not always least to greatest in this order
- depends on curve (increasing or decreasing)
-you can choose which to use or are told
-sometimes called circumscribed, or inscribed rectangles

Programs on Calculator RSUM
N = number of rectangles desired

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Definite Integrals

-we looked at sums of areas of rectangles
-leads to definite integrals

-added area of rectangles
k = number of rectangles
Ck = midpoint, left or right
-what part of rectangle is top
DeltaXk = width of rectangle

1. Rectangles (# partitions) can be different sizes
-tend toward same size, easier to read and keep track of
2. If this sum exits the function can be integrated
-the value is the definite integral
Riemann Sum - the limit of the sum, making smaller widths

Or

3. This brings us to notation - Greek to Roman letters

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Ex. The interval [-1,3] is partitioned into n subintervals of equal
length. Let mk denote the midpoint of those intervals. Express
the limit as an integral

Ex. Evaluate

If f(x) is negative integrals are more exciting

Area is supposed to be positive
-yet all height of rectangles
would be negative
-giving negative area

When f(x)<0

Some functions include both positive and negative values
-Therefore will need to do a combination of both ideas

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-Therefore will need to do a combination of both ideas
Area above x-axis - Area below x-axis

Exploration: We know
Find the values of the following.

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If f(x)=c is constant on [a,b], then

Integrals on Calculator - just does a sum of rectangles

fnInt( f(x), x, a, b )

On AP test must see mathematical notation, not calc notation

Need to watch for discontinuous functions - problems insue

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General Rules of Definite Integrals
- Thinking of the rectangles, these rules make sense
- Similar to limit and derivative rules

Ex. Suppose                   Find the following:

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Ex. Show that the value of
Is less than 3/2

Average (Mean) Value of a function on [a,b]

Ex. Find the average value of
On [0,3]

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This is also the Mean Value THM for Definite Integrals
- the integral will take on this value at some point in the interval

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Differential to Integral Calculus
Exploration 2 (book): Suppose we are given the following function

y=f(x)

1. Copy the graph of f onto your own paper.
Chose any x greater than a in the interval [a,b] and mark it.
2. Using only vertical line segments, shade in the region
Between the graph of f and the x-axis from a to x.
3. Your shaded region represents a definite integral.
Explain why this integral can be written as

Why not

4. Compare your picture with others. Notice how your integral
Depends on where you chose x to be. The integral is therefore
A function of x. Call it F.
5. Recall the F'(x) is the limit of ΔF/Δx as Δx gets smaller.
Represent ΔF in your picture by drawing one more vertical
shading segment to the right of the last one you drew in step 2.
ΔF is the (signed) area of your segment.
6. Represent Δx in your picture by moving x to beneath your newly
drawn segment. That small change in Δx is the thickness of
7. What is now the height of your vertical segment?
8. Can you see why Newton and Leibniz could concluded that
F'(x) = f(x)?

Since we let

Then

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Then

So the integral is the anti-derivative!!

If F is any anti-derivative of f, then

Assume x = a then

Therefore

HUGE!!

Ex. Find

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Fundamental Theorem of Calculus
-we discovered both parts of this in the exploration

Part 1: If f is continuous on [a,b] then
Has a derivative at every point
s in [a,b] and

Find

Find

Find dy/dx if

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Find dy/dx if

Part 2: If f is continuous at every point in [a,b] and if F is any
anti-derivative of f on [a,b], then

Ex.

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Net Area vs. Area
The definite integral is the net area between the fn and x-axis
- if asking for area we need to process what is below
the x-axis so it can get a sign flip!!
**Graphing the curve is a great idea

Ex. Find the area between
And the x-axis on [0,3]

Ex. The fixed cost of starting a manufacturing run and producing
the first 10 units is \$200. After that, the marginal cost at x units
output is
Find the total cost of producing
the first 100 units.

Ex. Suppose a wholesaler receives a shipment of 1200 cases of
boxes of chocolates every 30 days. The chocolate is sold to
retailers at a steady rate, and x days after the shipment arrives,
the inventory of cases still on hand is

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Find the average daily inventory.

Also find the average daily holding cost if the cost of holding
one case is 3 cents per day.

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Trapezoidal Approximations

Area of a Trapezoid

Where

Ex. Use the trapezoid rule with n = 4 to estimate

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Aside -- Simpson's Rule uses parabolic arcs on top of rectangles
to approximate integrals
Most Accurate Method --Don't need to know how to do

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