applications of integration hanna kim & agatha wuh block a
TRANSCRIPT
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Applications of
IntegrationHanna Kim & Agatha Wuh
Block A
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Disk Method IntroDisk Method is in many
ways similar to cutting a rollcake
into infinitely thin pieces and
adding all of them together
to find out the volume of the whole.
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Delving into Disk Method
•Disk = right cylinder
•Volume of disk= (area of disk) * (width of disk)=πr2w
•Integration is simply an addition of infinitesimally thin disks in a given interval.
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Demonstration of how the whole volume is computed by the addition
of smaller ones
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Disk Method Integration
Horizontal Axis of Revolution
Vertical Axis of Revolution
V =π [R(x)]2a
b
∫ dx
V =π [R(y)]2c
d
∫ dy
πr2=Area of Circle
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y =sinx+ 3
π(sinx+ 3)2 dx−π2
3π2
∫
fnInt((π(sin(x) + 3)2 ),x,−π /2, (3π ) / 2 ≈187.522 units3
The area under the curve y is rotated about the x-axis in the interval [-π/2, 3π/2]. Find its volume.
Disk Method Example
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Washer Method Intro
Washer’s method is
an extension of disk method. However, the
difference is that there is a hole in
the middle just like a bamboo or a
collection of CDs!
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•Washer=a disk with a hole
•Using this method is to find out an area of washer by subtracting the smaller circle from the bigger circle and then adding all the washers up.
Delving into Washer Method
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Washer Method Integration
V =π [R(x)]2 −[r(x)a
b
∫ ]2dx
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Washer Method Example
The region bounded by y=-5(x-3)2+8 and y=sin x +2 is revolved around the x-axis. Find its volume.
y =−5(x−3)2 + 8y=sinx+ 2
Find points of intersection
x =1.991 and x=4.171
1. Graph the two equations.2. Press 2nd Trace or Calc3. Press 5: intersect4. Identify the two different graphs.5. Done!
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Washer Method Example
π R2 − r2
a
b
∫ dx
π [(1.991
4.171
∫ − 5(x − 3)2 + 8)2 − (sin x + 2)2 ]dx
fnInt((−5(x − 3)2 + 8)2 − (sin(x) + 2)2 , x,1.991,4.171)
≈ 75.713
Ans *π ≈ 237.860 units3
Outer circleInner circle
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Shell Method IntroShell Method is like
“unrolling” a toilet paper and adding up the
infinitely thin layers of paper.
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Delving into Shell Method
Volume of shell= volume of cylinder-volume of hole=2πrhw
2π(average radius)(height)(thickness)
2πrh=circumference*height
=Area of cylinder without the top and bottom circles of the cylinder
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Shell Method Integration
V =2π p(y)h(y)dyc
d
∫ V =2π p(x)h(x)dxa
b
∫
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Shell Method ExampleThe region bounded by the curve , the x-axis,
and the line x=9 is revolved about the x-axis to generate a solid. Find the volume of the solid.
y = x
(9,3)y = x
x=y2
2π(y)(9 −y2 )dy0
3
∫
2π 9y−y3 dy0
3
∫
2π[9y2
2−
y4
4 0
3]
2π[9 * 32
2−34
4]
81π2
radius heightThe integral is in terms
of y because y is parallel to the axis of
rotation.
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Shell Method Example
y = x
x=y2
2π(y)(9 −y2 )dy0
3
∫
2π 9y−y3 dy0
3
∫
2π[9y2
2−
y4
4 0
3]
2π[9 * 32
2−34
4]
81π2
Constant multiple rule
Evaluate from 0 to 3
Answerunits3
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Section Method Intro
The section method is like
cutting a loaf of bread into
different slices and adding
those slices up.
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•Utilizing simple area equation of the cross section.
•Formula is applicable to any cross section shape
•Most common cross sections are squares, rectangles, triangles, semicircles, trapezoids
Delving into Section Method
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Section Method Integration
V = A(x)dxa
b
∫
Cross section taken perpendicular to the x-
axis
Cross section taken perpendicular to the y-
axis
V = A(y)dya
b
∫
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Section Method Example
The solid lies between planes perpendicular to the y-axis at y=0 and y=2. The cross sections perpendicular
to the y-axis are half circles with diameters ranging from the y-axis to the parabola
x = 5y2
d= 5y2
r =52
y2
The integral is in terms of y because the
cross sections are
perpendicular to the y-axis.
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Section Method Example
1
2π(
52
y2 )2 dy0
2
∫12π(
54)y4 dy
0
2
∫58π y4 dy
0
2
∫58π[
y5
5 0
2]
58π[
25
5−05
5]
58π[
325]
4π
1
2π(
52
y2 )2 dy0
2
∫12π(
54)y4 dy
0
2
∫58π y4 dy
0
2
∫58π[
y5
5 0
2]
58π[
25
5−05
5]
58π[
325]
4πConstant multiple rule
1/2 πr2=half circleEvaluate
Simplify
units3 Answer
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Shell Method VS.
Disk Method
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How Can We Distinguish?•For disk method, the
rectangle (disk) is always perpendicular to the axis of revolution
•For shell method, the rectangle is always parallel to the axis of revolution.
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Works Cited
•http://library.thinkquest.org/3616/Calc/S3/TSM.html
•http://mathdemos.gcsu.edu/mathdemos/shellmethod/2curvesshells.gif
•http://mathdemos.gcsu.edu/mathdemos/sectionmethod/sectionmethod.html
•http://mathdemos.gcsu.edu/mathdemos/washermethod/