lesson b.2 (pg. a22) the area between two · pdf file1 lesson b.2 (pg. a22) the area between...

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LessonB.2(pg.A22)TheAreaBetweenTwoCurves
Fromthislesson,youshouldbeableto:1.Findtheareabetweentwocurves2.Sketchthegraphs3.Setupnecessaryintegral(s)4.Findpointsofintersectionifnecessary5.Integrateandevaluate6.Solvewordproblemsinvolvingtheareabetweentwocurves

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Example:Findtheareabetweenand
Video:howtofindtheareabetweentwocurves: http://www.dummies.com/howto/content/howtofindtheareabetweentwocurves.html
Area=
Area=
http://www.dummies.com/how-to/content/how-to-find-the-area-between-two-curves.html

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HWFindtheareabetweenand1)
Findtheareabetweenand2)

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Findtheareabetweenand

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Example:Twoadvertisingagenciesarecompetingforamajorclient.TherateofchangeoftheclientsrevenuesusingAgencyAsadcampaignisapproximatedbyf(x)below.TherateofchangeoftheclientsrevenuesusingAgencyBsadcampaignisapproximatedbyg(x)below.Inbothcases,xrepresentstheamountspentonadvertising.Ineachcase,totalrevenueistheareaunderthecurve.
AgencyA AgencyB
Thisgraph(below)showstherelationshipbetweenthetworevenuefunctions.Weseethatonefunctionisabovetheother.Theareabetweenthetwofunctions(shaded)representstheadditionalrevenuethatwouldberealizedbyusingAgencyBsadcampaign.

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Example:Withoutanyefforttocurbpopulationgrowth,agovernmentestimatesthatitspopulationwillgrowattherateofthousandpeopleperyear.However,theybelievethataneducationprogramwillalterthegrowthratetothousandpeopleperyearoverthenext5years.Howmanyfewerpeoplewouldtherebeinthecountryiftheeducationprogramisimplementedandissuccessful?

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Withoutanyefforttocurbpopulationgrowth,agovernmentestimatesthatitspopulationwillgrowattherateofthousandpeopleperyear.However,theybelievethataneducationprogramwillalterthegrowthratetothousandpeopleperyearoverthenext5years.Howmanyfewerpeoplewouldtherebeinthecountryiftheeducationprogramisimplementedandissuccessful?
Xmin:8Xmax:8Xscl:1Ymin:3Ymax:100Yscl:10

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eIntegrationFormulas:

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Withoutanyefforttocurbpopulationgrowth,agovernmentestimatesthatitspopulationwillgrowattherateofthousandpeopleperyear.However,theybelievethataneducationprogramwillalterthegrowthratetothousandpeopleperyearoverthenext5years.Howmanyfewerpeoplewouldtherebeinthecountryiftheeducationprogramisimplementedandissuccessful?
t
p

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Example:Aconsumermagazinetestedtwokindsofengines.Onewasastandardengine,anditwasdeterminedthatitsaccelerationcouldbemodeledbyf(t)=6+.7tfeetpersec2,tsecondsafterstartingfromrest.Theaccelerationoftheturbochargedmodelcouldbeapproximatedbyg(t)=6+1.6t+.05t2feetpersec2,tsecondsafterstartingfromrest.Howmuchfasteristheturbochargedmodelmovingthanthestandardmodelattheendofa10secondtrial?

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Example:Findtheareaoftheface(notincludingthemouthandeyes)inthedrawingbelow.
Thelargecirclehasradius3andiscenteredatthepoint(0,2).Themouthistheareaboundedbythecurvesy=x2andy=0.5x2+0.5.Theeyesarecirclesofradius0.5centeredatthepoints(1,2.5)and(1,2.5).

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http://archives.math.utk.edu/visual.calculus/5/area2curves.3/index.htmlWebsiteswithexampleproblemsfindingtheareabetweencurves:
http://www.ltcconline.net/greenl/courses/106/AreaVolume/area2cur.htm
http://www.schooltube.com/video/2e5228080098b3c2e865/AreaBetweenTwoCurvesVideosexplaininghowtofindtheareabetweentwocurves:
http://www.youtube.com/watch?v=dK6sCvA0OJU
PowerPointpresentationofareabetweentwocurves:
AreaBetweenTwoCurves.ppt
http://archives.math.utk.edu/visual.calculus/5/area2curves.3/index.htmlhttp://www.ltcconline.net/greenl/courses/106/areavolume/area2cur.htmhttp://www.schooltube.com/video/2e5228080098b3c2e865/area-between-two-curveshttp://www.youtube.com/watch?v=dk6scva0oju
The Area Between Two Curves
Lesson 6.1
What If ?
We want to find the area between
f(x) and g(x) ?
Any ideas?
When f(x) < 0
Consider taking the definite integral for the function shown below.
The integral gives a negative area (!?)
We need to think of this in a different way
a
b
f(x)
Another Problem
What about the area between the curve and the x-axis for y = x3
What do you get for
the integral?
Since this makes no sense we need another way to look at it
Solution
We can use one of the properties of integrals
We will integrate separately for
-2 < x < 0 and 0 < x < 2
General Solution
When determining the area between a function and the x-axis
Graph the function first
Note the zeros of the function
Split the function into portions
where f(x) > 0 and f(x) < 0
Where f(x) < 0, take
absolute value of the
definite integral
Try This!
Find the area between the function
h(x)=x2 + x 6 and the x-axis
Note that we are not given the limits of integration
We must determine zeros
to find limits
Also must take absolute
value of the integral since
specified interval has f(x) < 0
Area Between Two Curves
Consider the region between
f(x) = x2 4 and g(x) = 8 2x2
Must graph to determine limits
Now consider function inside
integral
Height of a slice is g(x) f(x)
So the integral is
The Area of a Shark Fin
Consider the region enclosed by
Again, we must split the region into two parts
0 < x < 1 and 1 < x < 9
Slicing the Shark the Other Way
We could make these graphs as functions of y
Now each slice is
y by (k(y) j(y))
Practice
Determine the region bounded between the given curves
Find the area of the region
Horizontal Slices
Given these two equations, determine the area of the region bounded by the two curves
Note they are x in terms of y
Assignments A
Lesson 7.1A
Page 452
Exercises 1 45 EOO
Integration as an Accumulation Process
Consider the area under the curve y = sin x
Think of integrating as an accumulation of the areas of the rectangles from 0 to b
b
Integration as an Accumulation Process
We can think of this as a function of b
This gives us the accumulated area under the curve on the interval [0, b]
Try It Out
Find the accumulation function for
Evaluate
F(0)
F(4)
F(6)
Applications
The surface of a machine part is the region between the graphs of y1 = |x| and
y2 = 0.08x2 +k
Determine the value for k if the two functions are tangent to one another
Find the area of the surface of the machine part
Assignments B
Lesson 7.1B
Page 453
Exercises 57 65 odd, 85, 88
()
b
a
fxdx
2
3
2
xdx
-
()()()
bcb
aac
fxdxfxdxfxdx
=+
202
333
220
xdxxdxxdx
--
=+
[
]
2
2
()()
gxfxdx
-
-
()99()9
fxxgxxxaxis
=-=--
(
)
22
1
()9()9
9
jyxyandkyxy
==-==-
[
]
3
0
()()
kyjydy
-
2
6
yxyx
==-
2
2
8
xy
xy
=-
=
0
sin
b
xdx
0
0
()sincos()cos1
b
b
Abxdxxb
==-=-+
2
0
1
()2
2
x
Fxtdt
=+
SMART Notebook

Attachments
AreaBetweenTwoCurves.ppt
The Area Between Two Curves
Lesson 6.1
What If ?
We want to find the area between
f(x) and g(x) ?
Any ideas?
When f(x) < 0
Consider taking the definite integral for the function shown below.
The integral gives a negative area (!?)
We need to think of this in a different way
a
b
f(x)
Another Problem
What about the area between the curve and the x-axis for y = x3
What do you get for
the integral?
Since this makes no sense we need another way to look at it
Solution
We can use one of the properties of integrals
We will integrate separately for
-2 < x < 0 and 0 < x < 2
General Solution
When determining the area between a function and the x-axis
Graph the function first
Note the zeros of the function
Split the function into portions
where f(x) > 0 and f(x) < 0
Where f(x) < 0, take
absolute value of the
definite integral
Try This!
Find the area between the function
h(x)=x2 + x 6 and the x-axis
Note that we are not given the limits of integration
We must determine zeros
to find limits
Also must take absolute
value of the integral since
specified interval has f(x) < 0
Area Between Two Curves
Consider the region between
f(x) = x2 4 and g(x) = 8 2x2
Must graph to determine limits
Now consider function inside
integral
Height of a slice is g(x) f(x)
So the integral is
The Area of a Shark Fin
Consider the region enclosed by
Again, we must split the region into two parts
0 < x < 1 and 1 < x < 9
Slicing the Shark the Other Way
We could make these graphs as functions of y
Now each slice is
y by (k(y) j(y))
Practice
Determine the region bounded between

lesson b.2 (pg. a22) the area between two · pdf file1 lesson b.2 (pg. a22) the area between...

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