מקצת מעשה התורה a.k.a. The Halakhic Letter a.k.a. MMT a.k.a. 4Q394-399.
7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing.
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7.3.3 Volume by Cross- sectional Areas A.K.A. - Slicing
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Transcript of 7.3.3 Volume by Cross-sectional Areas A.K.A. - Slicing.
7.3.3 Volume by Cross-sectional Areas
A.K.A. - Slicing
I. Slicing
It is possible to find the volume of a solid (not necessarily a SOR) by integration techniques if parallel cross-sections obtained by slicing solid with parallel planes perpendicular to an axis have the same basic shape.
If the area of a cross-section is known and can be expressed in terms of x or y, then the area of a typical slice can be determined. The volume can be obtained by letting the number of slices increase indefinitely.
Therefore,
1
lim ( ) ( )bn
in
i a
A x x A x dx
II. Examples
A.) Assume that the base of a solid is the circle
and on each chord of the circle parallel to the y-axis there is erected a square. Find the volume of the resulting solid.
2 2 9x y
2
-2
BASE
2 2 9x y
2, 9x x
CROSS-SECTION
2, 9x x 22 9 x
22 9 x
Volume
( )b
a
A x dx
3
2
3
4 9 x dx
3
2
3
36 4x dx
144 cu. units
33
3
436
3x x
3 22
3
2 9 x dx
B.) Find the volume of the solid whose base is the
region in the first quadrant bounded by
, the x-axis, and the y-axis, and whose
cross-sections taken perp. to the x-axis are
squares.
2
14
xy
1
2
BASE
2
14
xy
2
,14
xx
CROSS-SECTION
,0x2
14
x
2
14
x
Volume
( )b
a
A x dx
2 2 4
0
12 16
x xdx
16 cu. units
15
23 5
06 80
x xx
22 2
0
14
xdx
C.) Find the volume of the solid whose base is
between one arc of y = sin x and the x-axis, and
whose cross-sections perp. to the x-axis are
equilateral triangles.
1
-1
2
BASE
siny x
,sinx x
CROSS-SECTION
,0x
sin x
sin xsin x
3 sin
2
x
Volume
( )b
a
A x dx
2
0
3 sin
4
xdx
3 cu. units
8
0
3 sin 2
4 2 4
x x
0
1 3 sinsin
2 2
xx dx
0
3 1 cos 2
4 2 2
xdx
III. Other Links http://mathdemos.gcsu.edu/mathdemos/
sectionmethod/sectiongallery.html http://www.ies.co.jp/math/java/calc/index.html http://www.geocities.com/
pkving4math2tor7/7_app_of_the_intgrl/7_03_01_finding_vol_by_slicing.htm
http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_02_finding_vol_by_using_cylind_shells.htm
