Volume By Slicing
(Section 6.1)
We used definite integrals to compute areas by making slices
(rectangles) and adding up the areas of the slices.
For volumes we
will slice the solid into pieces whose volume can be approximated
and then added together.
Recall: Volume =
So, as long as we know the __________ of the base of a region,
we can find the volume by the equation ___________________.
DEF
The
volume
of a solid
of known integrable crosssectional
area
A(x)
from
x=a
to
x=b
is the integral of
A
from
a
to
b
,
__________________________
NOTE:
Always integrate along the axis perpendicular to a slice.
Steps to Calculate the Volume of a Solid:
1.
Sketch the solid and a typical crosssection
2.
Find a formula for
A(x)
, the area of a typical crosssection.
3.
Find the limits of integration
4.
Integrate
A(x)
using the FTC (part 2).
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A solid has slices perpendicular to the yaxis that are squares
with one edge in the xy plane.
The intersection of the solid with
the xy plane is the region between the curves x = y
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 Spring '08
 LLHanks
 Calculus, Definite Integrals, Integrals, Angles, xy plane

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