Graph Polynomials and Graph Transformations in Algebraic Graph
The graph: Similarly, parisa yazdjerdi.
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Transcript of The graph: Similarly, parisa yazdjerdi.
![Page 1: The graph: Similarly, parisa yazdjerdi.](https://reader030.fdocuments.us/reader030/viewer/2022033106/56649dbc5503460f94aaeadb/html5/thumbnails/1.jpg)
Application of Integral
Done by :
Fatma Al-nuaimi
Kashaf Bakali
Parisa Yazdjerdi
Heba Hammud
Nadine Bleibel
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Definition of Area by Integral
by : Fatma Al-nuaimi(201004421)
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Finding areas by integration.
Using Riemann sum.
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Use Riemann sum to find the value of:
The graph:
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Step 1:
We can determine the value by subdividing the region into rectangle:
When the number of rectangles n
The area of rectangle is A=L*W
Width=W= 1/n
Length= 1+(1/n)i
So,
i
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Step 3: Performing some algebraic manipulation:
i
i
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Step 4:
Taking the limit to calculate the area:
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Area under the curveBy: Kashaf Bakali
(201105803)
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Find the Area enclosed by the parabola and above the x-axis.
As the area to be calculated should be above x-axis so, .
We first find the points of intersection by solving both equations simultaneously. i.e.
So,
Hence, the intersection points are; .
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Now, we sketch the graph.
The graph tells us the limit.
In this case, we have to find area from -1 till 3.
So, dx
)
=
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Similarly,
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Find the area bounded between and
Firstly, we would find the intersection points by solving both given equations simultaneously. i.e.
We get the intersections points as,
We sketch the graph.
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Now we have, . We use the formula, In this case,
and. Hence,
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Area between two curveBy: Parisa Yazdjerdi
(201005599)
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Areas between two curves
Process of finding area between two curves consist of 3 main steps :
1. finding intersection of the curves ( put two equation in an equality)
2. Drawing the graph to distinguish intervals and exact areas
3. Using integral formula
parisa yazdjerdi
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Area between two curves
Example : find area between f(x) = Sin x , g(x) = Cos x , x = 0 and x = π/2 .
First step : find intersections
parisa yazdjerdi
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Area between tow curves
Second step : Drawing the Graph
parisa yazdjerdi
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Area between two curves
Third step : Using formula to find area
parisa yazdjerdi
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Area between two curves
parisa yazdjerdi
22222
4
)]2
1
2
1()10[()]10()
2
1
2
1[(
]sincos[]cos[sin
]cos[sin]sin[cos
so,
cosx -sinx|=cosx-sinx| so, 0>cosx -sinx
and
sinx-cosx|=sinx-cosx|so, 0 >sinx-cosx therefore24
cossin4
0;sincos
2
4
40
2
4
4
0
xxxx
dxxxdxxxA
xxxandxxx
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Volume of SolidBy : Hebba Hammud
(201003247)
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Volume of Solids
For any Solid(S),we cut it into pieces and approximate each piece by a cylinder. This is called : cross-sectional area.
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Definition of Volume
)(
)()(lim1
abAAdx
dxxAxxiAv
b
a
b
a
n
in
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Exercises(about the x-axis)
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Find the volume v resulting from the revolution of the region bounded by:y=√x , from x=0 to x=1 about the x-axis.
2)0
2
1(
2
)(
10
2
1
0
1
0
2
x
xdx
dxxv
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Find the volume v resulting from the revolution of the region bounded by:y=√(a2-x2 ) from x=-a to x=a and the x-axis about the
x-axis.
3
4]
3
2[2
]0)3
[(2
)3
(2
)(2
)(
)(
33
32
0
32
0
22
22
222
aa
aaa
xxa
dxxa
dxxa
dxxav
a
a
a
a
a
a
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Exercises (about the y-axis)
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Find the volume of the solid of revolution generated by rotating the curve y = x3 between y = 0 and y = 4 about the y-axis.
We first must express x in terms of y, so that we can apply the formula.
If y = x3 then x = y1/3
The formula requires x2, so x2 = y2/3
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Find the volume generated by the areas bounded by the Given curves if they are revolved about the y-
axis: y2 = x, y = 4 and x = 0 [revolved about the y-axis]
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VolumeBy: Nadine Bleibel
(201104593)
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Basics of a Cylinder
Nadine Bleibel
A cylinder is a simple solid which is boundedby a plane region B1- which is called the base.A cylinder also has a congruent region B2 in a Parallel plane.
The formula for volume for a circular cylinder isV=(Pi)r^2(h)
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EXAMPLE 1 (example 2 p 356)
Nadine Bleibel2
)02
1(
2
)(
10
2
1
0
1
0
2
x
xdx
dxxv
Find the Volume of the solid obtained by rotatingAbout the x-axis the region under the curvey= √x from 0 to 1.
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EXAMPLE 2 (example 3 p357)
Nadine Bleibel
EXAMPLE: Find the volume of the solid obtaining by rotating about the y-axis the region bounded by y = x3, y = 8, and x = 0.
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