Calculus 7.3 Day 1

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7.3 Part One Volumes by Slicing
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Find the volume of the pyramid
Consider a horizontal slice through the pyramid.
The volume of the slice is s2dh.
If we put zero at the top of the pyramid and make
down the positive direction, then sh.
This correlates with the formula
s
dh
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Method of Slicing
Sketch the solid and a typical cross section.
Find a formula for V(x). (Note that I used V(x)
instead of A(x).)
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Find the limits of integration.
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Integrate V(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
p