Calculus
Lesson
34
Volume
- Discs and Cross-Sections
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First DISC Method
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|
1. Region is enclosed by
and
.
Rotate the region about the x-axis.
2. Region is bounded by
and
.
Almost the same picture
Rotate about
.
as above....
3. Region is enclosed by
and
. Rotate about x-axis.
4. Region is enclosed by
,
,
,
.
Rotate about y-axis.
Set up as two volumes, annulus for 1 < y < 2, disc for 0 < y < 1.
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Method:
1. Draw sketch
2. Identify endpoints
3. Identify whether to use
or
.
4. Draw picture of cross-section.
5. Write out
and
continue for either disc or annulus
6. Set up integral.
7. Evaluate integral
More Examples: you can do each of the above, only rotating about the other axis.
Also: Find the volume of a sphere (5 inches in radius)
with a 3-inch radius hole drilled through it.
(answer:
cubic inches)
For the CROSS SECTION Method:
Example: The base of a solid is
defined by
from
to
,
and each
cross section is a square perpendicular to the y-axis.
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We still use
,
but now our
cross
section is actually easier! Note that when
we draw our picture, we could draw
2D or 3D, whichever is easier for us.
Repeat example for perpendicular to x-axis, then for equilateral triangles,
then for semi-circles.
Back to Dr. Nandor's Calculus Notes Page
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