6.1 Area between two curves

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6.1 Area between two curves
??Ak area of k th rectangle,?f(ck) g(ck )
height, ??xk width.
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Find the area of the region between the curves
Figure 4.23 When the formula for a bounding
curve changes, the area integral changes to
match. (Example 5)
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Section 6.2 Figure 5
Approximating the volume of a sphere with radius 1
(b) Using 10 disks, V 4.2097
(c) Using 20 disks, V 4.1940
(a) Using 5 disks, V 4.2726
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6. 2 Volumes Solid of revolution
Figure 5.6 The region (a) and solid (b) in
Example 4.
y f(x) is rotated about x-axis on a,b. Find
the volume of the solid generated. A
cross-sectional slice is a circle and a slice is
a disk.
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Volumes Solid of revolution
Figure 5.6 The region (a) and solid (b) in
Example 4.
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Volumes by disk-y axis rotation
Find the volume of the solid generated by
revolving a region between the y-axis and the
curve x 2/y from y 1 to y 4.
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Find the volume of the solid generated by
revolving a region between the y-axis and the
curve x 2/y from y 1 to 4.
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Washers
Figure 5.10 The cross sections of the solid of
revolution generated here are washers, not disks,
so the integral ????A(x) dx leads to a slightly
different formula.
If the region revolved does not border on or
cross the axis of revolution, the solid has a
hole in it. The cross sections perpendicular to
the axis are washers.
b a
V Outside Volume Inside Volume
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. The region bounded by the curve y x2 1 and
the line y -x 3 is revolved about the x-axis
to generate a solid. Find the volume of the solid
of revolution.
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The inner and outer radii of the washer swept out
by one slice. Outer radius R - x 3 and the
inner radius r x2 1
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The inner and outer radii of the washer swept out
by one slice. Outer radius R - x 3 and the
inner radius r x2 1
Find the limits of integration by finding the
x-coordinates of the points of intersection.
x2 1 - x 3
x2 x 20
( x 2 )(x 1) 0
x -2 x 1
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Calculation of volume
Outer radius R - x 3 and the inner radius r
x2 1
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y-axis rotation
The region bounded by the parabola y x2 and the
line y 2x in the first quadrant is revolved
about the y-axis to generate a solid. Find the
volume of the solid.
Drawing indicates a dy integration so solve each
equation for x as a function of y
Set to find y limits of integration
y 0 and y 4 are limits
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The washer swept out by one slice perpendicular
to the y-axis.
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calculation
The region bounded by the parabola y x2 and the
line y 2x in the first quadrant is revolved
about the y-axis to generate a solid. Find the
volume of the solid.
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6. 3 Cylindrical Shells
Figure 5.17 Cutting the solid into thin
cylindrical slices, working from the inside out.
Each slice occurs at some xk between 0 and 3 and
has thickness ? x. (Example 1)
Used to find volume of a solid of revolution by
summing volumes of thin cylindrical shells or
sleeves or tree rings.
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volume of a shell
Imagine cutting and unrolling a cylindrical shell
to get a (nearly) flat rectangular solid. Its
volume is approximately V length ? height ?
thickness.
)
Vshell 2?(radius)(height)(thickness)
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problem
The region enclosed by the x-axis and the
parabola y f(x) 3x x2 is revolved about
the y axis. Find the volume of the solid of
revolution.
Vshell 2?(radius)(height)(thickness)
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The shell swept out by the kth rectangle.
Notice this axis or revolution is parallel to the
red rectangle drawn.
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problem
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The region, shell dimensions, and interval of
integration in
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The shell swept out by the rectangle in.
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Summary-Volumes-which method is best
Axis of rotation
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Lengths of Plane curves
Find the length of the arc formed by
u 1 4x du 4dx du/4 dx
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Follow the link to the slide. Then click on the
figure to play the animation.
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Figure 6.2.5
Figure 6.3.7
Figure 6.2.12
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Section 6.3 Figures 3, 4
Volumes by Cylindrical Shells
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Computer-generated picture of the solid in
Example 9
Section 1 / Figure 1
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