Calculus 7.3 Day 2

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7.3
VOLUMES
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Solids with Known Cross Sections

• If A(x) is the area of a cross section of a solid
  and A(x) is continuous on a, b, then the volume
  of the solid from x a to x b is

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Known Cross Sections

• Ex The base of a solid is the region enclosed
  by the ellipse
• The cross sections are perpendicular to the
  x-axis and are isosceles right triangles whose
  hypotenuses are on the ellipse. Find the volume
  of the solid.

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1.) Find the area of the cross section A(x).
y
2.) Set up evaluate the integral.
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Unknown Cross Sections DISC METHOD
Sketch the solid and a typical cross section.
Find a formula for A(x).
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Find the limits of integration.
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Integrate A(x) to find volume.
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A 45o wedge is cut from a cylinder of radius 3 as
shown.
Find the volume of the wedge.
You could slice this wedge shape several ways,
but the simplest cross section is a rectangle.
Since the wedge is cut at a 45o angle
Since
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Even though we started with a cylinder, p does
not enter the calculation!
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Cavalieris Theorem
Two solids with equal altitudes and identical
parallel cross sections have the same volume.
Identical Cross Sections
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Suppose I start with this curve.
My boss at the ACME Rocket Company has assigned
me to build a nose cone in this shape.
So I put a piece of wood in a lathe and turn it
to a shape to match the curve.
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How could we find the volume of the cone?
One way would be to cut it into a series of thin
slices (flat cylinders) and add their volumes.
In this case
r the y value of the function
thickness a small change in x dx
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If we add the volumes, we get
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Math Demo
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We use a horizontal disk.
The thickness is dy.
volume of disk
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The natural draft cooling tower shown at left is
about 500 feet high and its shape can be
approximated by the graph of this equation
revolved about the y-axis
The volume can be calculated using the disk
method with a horizontal disk.
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The region bounded by and
is revolved about the y-axis. Find the volume.
If we use a horizontal slice
The disk now has a hole in it, making it a
washer.
outer radius
inner radius
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This application of the method of slicing is
called the washer method. The shape of the slice
is a circle with a hole in it, so we subtract the
area of the inner circle from the area of the
outer circle.
Math Demo
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If the same region is rotated about the line x2
The outer radius is
The inner radius is
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We can use the washer method if we split it into
two parts
cylinder
inner radius
outer radius
thickness of slice
Japanese Spider Crab Georgia Aquarium, Atlanta
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cross section
If we take a vertical slice
and revolve it about the y-axis
we get a cylinder.
If we add all of the cylinders together, we can
reconstruct the original object.
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cross section
The volume of a thin, hollow cylinder is given by
r is the x value of the function.
h is the y value of the function.
thickness is dx.
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cross section
If we add all the cylinders from the smallest to
the largest
Math Demo
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Find the volume generated when this shape is
revolved about the y axis.
We cant solve for x, so we cant use a
horizontal slice directly.
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If we take a vertical slice
and revolve it about the y-axis
we get a cylinder.
Shell method
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• To find surface area, we can slice a solid and
  approximate the surface area of these slices by
  2p ? f(x) ? ?s, where ?s is the slant height of
  the slice.

• We will see in Section 7.4 that ?s can be written
  as

• To find surface area, use

(SA will exist if f and f are continuous on a,
b