x2 t06 03 parallel crosssections
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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
z
b
e.g. Find the volume of this rectangular pyramid
h
Volumes By Non Circular Cross-Sections
y
xa
z
y2
z
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
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
b21
b21
Find x in terms of z
x
z
h
b21
b21
Find x in terms of z
x
z
h
b21
b21
x
Find x in terms of z
x
z
h
b21
b21
x (x,z)
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
(0,h)
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
(0,h)
0,
21
b
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
bh
b
hm
221
0
0
(0,h)
0,
21
b
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
bh
b
hm
221
0
0
02 xb
hhz
(0,h)
0,
21
b
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
bh
b
hm
221
0
0
02 xb
hhz
hzhb
x
hxhzb
2
2
(0,h)
0,
21
b
Find x in terms of z
x
z
h
b21
b21
x (x,z)
ssimilar using :1 Method
h
b21
h - z
x
xzh
b
h
21
h
zhbx
xzh
bh
2
2
geometry coordinate using :2 Method
bh
b
hm
221
0
0
02 xb
hhz
hzhb
x
hxhzb
2
2
h
zhay
2
Similarly;
(0,h)
0,
21
b
hzha
hzhb
zA22
4
hzha
hzhb
zA22
4
2
2
hzhab
hzha
hzhb
zA22
4
2
2
hzhab
z
hzhab
V 2
2
hzha
hzhb
zA22
4
2
2
hzhab
z
hzhab
V 2
2
h
zz
zh
zhabV
02
2
0lim
hzha
hzhb
zA22
4
2
2
hzhab
z
hzhab
V 2
2
h
zz
zh
zhabV
02
2
0lim
h
dzzhhab
0
2
2
hzha
hzhb
zA22
4
2
2
hzhab
z
hzhab
V 2
2
h
zz
zh
zhabV
02
2
0lim
h
dzzhhab
0
2
2
h
zhhab
0
3
2 3
hzha
hzhb
zA22
4
2
2
hzhab
z
hzhab
V 2
2
h
zz
zh
zhabV
02
2
0lim
h
dzzhhab
0
2
2
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
xx
xxV
(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
xx
xxV
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
xx
xxV
2
0
248 dxx
2
0
3
31
48
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
xx
xxV
2
0
248 dxx
2
0
3
31
48
xx
3units 3
128
038
88
(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
xx
xxV
2
0
248 dxx
2
0
3
31
48
xx
3units 3
128
038
88
Exercise 3A; 16ade, 19
Exercise 3C; 2, 7, 8
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