Why$would$someone$be$interested$in$the$area$ · PDF...

Montana State University M161: Survey of Calculus 116

S. Schaefer

Section 6.6 -‐ Area Between Two Curves

Why would someone be interested in the area between two curves?

The area under the upper (red) curve represents the total acreage of corn planted over the last 20 years. The area under the lower (green) curve represents the total acreage of corn harvested over the last 20 years. The area between the curves represents the amount of acres that were planted, but not harvested, over the past 20 years.

The area under the upper (blue) curve represents the total kWh usage per Capita in the US from 1960 to 2006 The area under the lower (yellow) curve represents the total kWh usage per Capita in CA from 1960 to 2006 The area between the curves represents the amount of electricity saved after new regulations were put into effect in CA, from 1974 to 2006, when compared to kWh per Capita in the US.

The area under the upper (blue) curve represents the total growth of employment in the healthcare field over the past 10 yrs The area under the lower (red) curve represents the total growth of employment in all other industries over the past 10 years The area between the graphs represents how many more new jobs there were in healthcare than in all the other fields.


S. Schaefer

Finding the Area Between Two Curves

!"# ! ! !"# ! ! !" !"#$%#&"&' !" !, ! !"#ℎ !ℎ!" ! ! ≥ ! ! !"# !"!#$ ! !" !ℎ! !"#$%&'( [!, !] Then the area of the region

bounded ABOVE by y = f(x) and bounded BELOW by y = g(x)

on [a, b] is given by:

!"#$ = ! ! − !(!) !"!

!

The area under each curve represents how much snow water equivalent (in inches) accumulated over that year. The area under the green curve gives the snow water equivalent on average. The area between the curve for the average and any of the other curves represents the difference between that year and the average in snow water equivalent.

The area under f(t) represents the rate a country’s petroleum consumption is expected to grow (in millions of barrels per year) over the next 5 years. The area under g(t) represents the rate a country’s petroleum consumptions is expected to grow after the country implements energy-‐conservations measures. The area between the curves represents the amount of petroleum that would be saved over the 5-‐year period because of the conservation measures.


S. Schaefer

Area between Two Curves The BUSI Method

!"#$ = !""#$ !"#$%&'# − !"#$% !"#$%&'# !"!

! Ex. Area between

! = !! !"# ! = 2! − !!

Step B: Find Bounds & Intersections Case 1: BOUNDS are given

Check for intersections between bounds Case 2: NO BOUNDS are given

Use all intersections found Finding INTERSECTIONS

Set !"#$%&'# 1 = !"#$%&'# 2 Solve for x.

Step U: Find Upper & Lower Function on each interval

Create Intervals from Step B values Plug test point into both functions

Largest value = Upper Function Smallest value = Lower Function

Step S: Subtract !""#$ !"#$%&'# − !"#$% !"#$%&'#

Remember to distribute the subtraction sign Step I: Integrate

Integrate Step S, Use interval endpoints as limits for the integral If there is more than one interval Integrate over each interval and add then up

Step B !! = 2! − !! 2!! − 2! = 0 2!(! − 1) = 0

! = 0, 1

Step U Interval (0,1)

Test Value 12

! = !! 12

!=14

! = 2! − !! 212−

12

!=34

Upper Function 2! − !!

Lower Function !!

Step S 2! − 2!!

Step I

!"#$ = (2! − !!)!

!!"

= !! −2!!

310=!!


S. Schaefer

Examples #1 Intersections: One Region Find the area of the region bounded by ! = 5! − !! !"# ! = !

Step B Bounds & Intersections Any bounds given? NO or YES ________________________ Intersections:

Step I Integrate

Step U Upper and Lower Functions

Interval/Region

Test Value

Function 1: ! = 5! − !!

Function 2: ! = !

Upper Function

Lower Function

Step S Subtract (Upper) – (Lower)


S. Schaefer

#2 Intersections: Two Regions Find the area of the regions bounded by ! = !! − ! !"# ! = 3!


Step I Integrate


Interval/Region

Test Value

Function 1: ! = !! − !

Function 2: ! = 3!

Upper Function

Lower Function



S. Schaefer

#3 Bounds: One Region Find the area of the region bounded by: ! = !! − 2! !"# ! = ! + 4 & !"#$% ! = 1 !"# ! = 3


Step I Integrate


Interval/Region

Test Value

Function 1: ! = !! − 2!

Function 2: ! = ! + 4

Upper Function

Lower Function



S. Schaefer

#4 Intersections VS Bounds: A Comparison #4a: INTERSECTIONS Find the area enclosed by the graphs of: ! = 2! − 1 !"# ! = !! − 4

#4b: BOUNDS Find the area enclosed by the graphs of:

! = 2! − 1 !"# ! = !! − 4 !"# !ℎ! !"#$% ! = 1 !"# ! = 2

Step B Bounds & Intersections Bounds given? NO or YES _____________ Intersections:

Step B Bounds & Intersections Bounds given? NO or YES _____________

#4a: INTERSECTIONS Step U

Upper & Lower Functions

#4b: BOUNDS

Interval/Region

Test Value

Function 1: ! = 2! − 1

Function 2: ! = !! − 4

Upper Function

Lower Function


Step I Integrate

Step I Integrate


S. Schaefer

#5 Intersections AND Bounds: Three Regions Find the area of the regions bounded by ! = 8− !! !"# ! = !! !"# !ℎ! !"#$% ! = −3 !"# ! = 3


Step I Integrate


Interval/Region

Test Value

Function 1: ! = 8− !!

Function 2: ! = !!

Upper Function

Lower Function


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