Difference between revisions of "Math 22 The Area of a Region Bounded by Two Graphs"

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   and the Producer Surplus is <math>PS=\int_0^{x_0} [p_0-s(x)]dx</math>
 
   and the Producer Surplus is <math>PS=\int_0^{x_0} [p_0-s(x)]dx</math>
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'''Exercises'''
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'''2)'''  Find the consumer and producer surpluses by using the demand <math>p=50-0.5x</math> and supply functions <math>p=0.125x</math>.
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|Find the solution (equilibrium point): <math>50-0.5x=0.125x</math>, so <math>0.625x=50</math>, so <math>x=80</math>, then <math>p(80)=0.125(80)=50-0.5(80)=10</math>. Therefor, <math>x_0=80</math> and <math>p_0=10</math>
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|So <math>CS=\int_0^{80} [50-0.5x-10]dx=\int_0^{80} [40-0.5x]dx=[40x-\frac{1}{4}x^2]\Biggr |_0^{80}=40(80)-\frac{1}{4}(80)^2=1600</math>
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|and <math>PS=\int_0^{80} [10-0.125x]dx=[10x-\frac{0.125}{2}x^2]\Biggr |_0^{80}=10(80)-\frac{0.125}{2}(80^2)=400</math>
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|}
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 08:28, 17 August 2020

Area of a Region Bounded by Two Graphs

Region 2 graph.png

 If  and  are continuous on  and  for all  in , 
 then the area of the region bounded by the graphs of  given by
 
 

Exercises

1) Find the area of the region bounded by the graph of and the graph of .

Solution:  
Find the bound of the region by setting , so , hence , then , therefore and
Check which function is the top function by choosing one number in between the bound and plug in:
Pick , so , and . Therefore, will be the top function.

Consumer Surplus and Producer Surplus

CS vs PS.png

 Given the demand function is  and the supply function is . 
 Let  be the solution of .
 
 Then, the Consumer Surplus is 
 
 and the Producer Surplus is 

Exercises

2) Find the consumer and producer surpluses by using the demand and supply functions .

Solution:  
Find the solution (equilibrium point): , so , so , then . Therefor, and
So
and

Return to Topics Page

This page were made by Tri Phan