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Montana State University M161: Survey of Calculus 116 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 Montana State University M161: Survey of Calculus 117 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. Finding the Area Between Two Curves !"# ! ! !"# ! ! !" !"#$%#&"&' !" !, ! !"#ℎ !ℎ!" ! ! ≥ ! ! !"# !"!#$ ! !" !ℎ! !"#$%&'( [!, !] Then the area of the region bounded ABOVE by y = f(x) and bounded BELOW by y = g(x) on [a, b] is given by: !
! ! − !(!) !" !"#$ =
!
S. Schaefer Montana State University M161: Survey of Calculus 118 Area between Two Curves The BUSI Method !
Ex. Area between !""#$ !"#$%&'# − !"#$% !"#$%&'# !" !"#$ =
! = ! ! !"# ! = 2! − ! ! !
Step B: Find Bounds & Intersections Step B 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. ! ! = 2! − ! ! 2! ! − 2! = 0 2!(! − 1) = 0 ! = 0, 1 Step U (0,1) Interval 1
2
Test Value 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 !""#$ !"#$%&'# − !"#$% !"#$%&'# Remember to distribute the subtraction sign 1
2
! = 2! − ! ! 1
1
2
−
2
2
Integrate Step S, Use interval endpoints as limits for the integral If there is more than one interval Integrate over each interval and add then up 1
= 4
2! − ! ! Lower Function ! ! Step I !
!"#$ =
(2! − ! ! ) !" !
2! ! 1 !
= 3 0 !
3
= 4
2! − 2! ! = !! −
!
Upper Function Step S Step I: Integrate !
! = ! ! S. Schaefer Montana State University M161: Survey of Calculus 119 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 U Upper and Lower Functions Interval/Region Test Value Function 1: !
! = 5! − ! Function 2: ! = ! Upper Function Lower Function Step S Subtract (Upper) – (Lower) Step I Integrate S. Schaefer Montana State University M161: Survey of Calculus 120 #2 Intersections: Two Regions Find the area of the regions bounded by ! = ! ! − ! !"# ! = 3! Step B Bounds & Intersections Any bounds given? NO or YES ________________________ Intersections: Step U Upper and Lower Functions Interval/Region Test Value Function 1: ! = ! ! − ! Function 2: ! = 3! Upper Function Lower Function Step S Subtract (Upper) – (Lower) Step I Integrate S. Schaefer Montana State University M161: Survey of Calculus 121 #3 Bounds: One Region Find the area of the region bounded by: ! = ! ! − 2! !"# ! = ! + 4 & !"#$% ! = 1 !"# ! = 3 Step B Bounds & Intersections Any bounds given? NO or YES ________________________ Intersections: Step U Upper and Lower Functions Interval/Region Test Value Function 1: !
! = ! − 2! Function 2: ! = ! + 4 Upper Function Lower Function Step S Subtract (Upper) – (Lower) Step I Integrate S. Schaefer Montana State University M161: Survey of Calculus 122 #4 Intersections VS Bounds: A Comparison #4a: INTERSECTIONS #4b: BOUNDS Find the area enclosed by the graphs of: ! = 2! − 1 !"# ! = ! ! − 4 Find the area enclosed by the graphs of: ! = 2! − 1 !"# ! = ! ! − 4 !"# !ℎ! !"#$% ! = 1 !"# ! = 2 Step B Bounds & Intersections Step B Bounds & Intersections Bounds given? NO or YES _____________ Intersections: #4a: INTERSECTIONS Bounds given? NO or YES _____________ Step U Upper & Lower Functions #4b: BOUNDS Interval/Region Test Value Function 1: ! = 2! − 1 Function 2: ! = ! ! − 4 Upper Function Lower Function Step S Subtract (Upper) – (Lower) Step I Integrate Step I Integrate S. Schaefer Montana State University M161: Survey of Calculus 123 #5 Intersections AND Bounds: Three Regions Find the area of the regions bounded by ! = 8 − ! ! !"# ! = ! ! !"# !ℎ! !"#$% ! = −3 !"# ! = 3 Step B Bounds & Intersections Any bounds given? NO or YES ________________________ Intersections: Step U Upper and Lower Functions Interval/Region Test Value Function 1: ! = 8 − ! ! Function 2: ! = ! ! Upper Function Lower Function Step S Subtract (Upper) – (Lower) Step I Integrate S. Schaefer