Integration

1
Integration

• Math 115B
• Fall 2003

2
Motivation

• Given a demand curve, graphically we know the
  relationship between demand and revenue

3
Consumer Surplus

• Looking at the graph, we see that if q units were
  sold at a price D(q), some of those people would
  have been willing to pay more than D(q).
• Graphically, this represents the region above the
  revenue and is called consumer surplus

4
Not Sold Revenue

• We also see that we lost revenue because some
  people would have bought the product at a lower
  unit price
• This region is to the right of the revenue

5
What do we want?

• We want to determine a way to calculate those
  revenues
• We already know how to calculate the revenue of
  the shaded region
• How do we calculate the consumer surplus and not
  sold revenue? 

6
Review

• If fX(x) is the p.d.f of a continuous random
  variable X with c.d.f FX(x) , then the area A is
  equal to
• P(a lt X lt b)P(a ? X ? b)FX(b) - FX(a)
• This translates graphically to the shaded region
  on the graph

7
Graphical Representation

• We would like to find the area of the shaded
  region
• This would allow us to find the area of the
  consumer surplus and not sold revenue
• Integration

8
More Practice

• Set up, but do not evaluate, an integral that
  gives the area between the graph of the function
  and the x-axis for x
  between 1 and 6.
• Set up, but do not evaluate, an integral that
  gives the area between the graph of
• and the x-axis for x between 0
  and 8.

9
Applications

• Now that we have looked at what integration is,
  we need to see what applications it for us
• i.e., how can it help us with Project 1?
• Applications

10
Connections

• Especially for Project 2, you will need to use
  Integration in relation to what you learned in
  Math 115A
• Probability Density Functions