# 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> | ||

+ | |||

+ | '''Exercises''' | ||

+ | |||

+ | '''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>. | ||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |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> | ||

+ | |- | ||

+ | |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> | ||

+ | |- | ||

+ | |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> | ||

+ | |} | ||

[[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

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: |
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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

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 |

**This page were made by Tri Phan**