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Volumes By Non Circular Cross-Sections
Volumes By Non Circular Cross-Sections
a
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
zb
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zb
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zx2
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zx2
b
e.g. Find the volume of this rectangular pyramid
h
xyzA 4
Find x in terms of z
x
z
h
b21b
21
Find x in terms of z
x
z
h
b21b
21
Find x in terms of z
x
z
h
b21b
21
x
Find x in terms of z
x
z
h
b21b
21
x (x,z)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
(0,h)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
(0,h)
0,
21 b
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
bh
b
hm
2210
0
(0,h)
0,
21 b
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
bh
b
hm
2210
0
02
x
bhhz
(0,h)
0,
21 b
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
bh
b
hm
2210
0
02
x
bhhz
hzhbx
hxhzb
2
2
(0,h)
0,
21 b
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometrycoordinateusing:2 Method
bh
b
hm
2210
0
02
x
bhhz
hzhbx
hxhzb
2
2
h
zhay2
Similarly;
(0,h)
0,
21 b
h
zhah
zhbzA22
4
h
zhah
zhbzA22
4
2
2
hzhab
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
h
dzzhhab
0
22
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
h
dzzhhab
0
22
h
zhhab
0
3
2 3
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
h
dzzhhab
0
22
h
zhhab
0
3
2 3
3
3
2
units 3
30
abh
hhab
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x-2
-2
2
2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
x
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
2
2
2
044lim
xxxxV
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
2
2
2
044lim
xxxxV
2
0
248 dxx
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
2
2
2
044lim
xxxxV
2
0
248 dxx
2
0
3
3148
xx
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
2
2
2
044lim
xxxxV
2
0
248 dxx
2
0
3
3148
xx
3units 3
128
03888
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
y
x2
2
-2
-2
xy2
y2
24yxA
244 x
xxV 244
2
2
2
044lim
xxxxV
2
0
248 dxx
2
0
3
3148
xx
3units 3
128
03888
Exercise 3A; 16ade, 19
Exercise 3C; 2, 7, 8