Definite Integral

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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

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                               Sub          0,1        1,2        2,3

                                 Sub          0,1           1,2    2,3

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


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


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
   your segment.
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


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So the integral is the anti-derivative!!

If F is any anti-derivative of f, then

Assume x = a then



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 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


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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


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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