x2 t05 03 parallel crosssections (2012)
TRANSCRIPT
![Page 1: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/1.jpg)
Volumes By Non Circular Cross-Sections
![Page 2: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/2.jpg)
Volumes By Non Circular Cross-Sections
a
b
e.g. Find the volume of this rectangular pyramid
h
![Page 3: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/3.jpg)
Volumes By Non Circular Cross-Sections
y
xa
b
e.g. Find the volume of this rectangular pyramid
h
![Page 4: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/4.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
b
e.g. Find the volume of this rectangular pyramid
h
![Page 5: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/5.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
b
e.g. Find the volume of this rectangular pyramid
h
![Page 6: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/6.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
zb
e.g. Find the volume of this rectangular pyramid
h
![Page 7: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/7.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zb
e.g. Find the volume of this rectangular pyramid
h
![Page 8: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/8.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zx2
b
e.g. Find the volume of this rectangular pyramid
h
![Page 9: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/9.jpg)
Volumes By Non Circular Cross-Sections
y
xa
z
y2
zx2
b
e.g. Find the volume of this rectangular pyramid
h
xyzA 4
![Page 10: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/10.jpg)
Find x in terms of z
x
z
h
b21b
21
![Page 11: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/11.jpg)
Find x in terms of z
x
z
h
b21b
21
![Page 12: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/12.jpg)
Find x in terms of z
x
z
h
b21b
21
x
![Page 13: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/13.jpg)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
![Page 14: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/14.jpg)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
![Page 15: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/15.jpg)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
![Page 16: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/16.jpg)
Find x in terms of z
x
z
h
b21b
21
x (x,z)
ssimilar using :1Method
h
b21
h - z
x
![Page 17: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/17.jpg)
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
![Page 18: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/18.jpg)
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
![Page 19: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/19.jpg)
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
![Page 20: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/20.jpg)
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)
![Page 21: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/21.jpg)
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
![Page 22: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/22.jpg)
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
![Page 23: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/23.jpg)
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
![Page 24: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/24.jpg)
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
![Page 25: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/25.jpg)
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
![Page 26: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/26.jpg)
h
zhah
zhbzA22
4
![Page 27: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/27.jpg)
h
zhah
zhbzA22
4
2
2
hzhab
![Page 28: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/28.jpg)
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
![Page 29: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/29.jpg)
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
![Page 30: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/30.jpg)
h
zhah
zhbzA22
4
2
2
hzhab
zh
zhabV
2
2
h
zzz
hzhabV
02
2
0lim
h
dzzhhab
0
22
![Page 31: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/31.jpg)
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
![Page 32: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/32.jpg)
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
![Page 33: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/33.jpg)
(ii) A solid has a circular base, , and each cross section is a square perpendicular to the base.
422 yx
![Page 34: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/34.jpg)
(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
![Page 35: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/35.jpg)
(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
![Page 36: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/36.jpg)
(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
![Page 37: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/37.jpg)
(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
![Page 38: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/38.jpg)
(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
![Page 39: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/39.jpg)
(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
![Page 40: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/40.jpg)
(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
![Page 41: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/41.jpg)
(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
![Page 42: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/42.jpg)
(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
![Page 43: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/43.jpg)
(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
![Page 44: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/44.jpg)
(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
![Page 45: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/45.jpg)
(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
![Page 46: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/46.jpg)
(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
![Page 47: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/47.jpg)
(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
![Page 48: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/48.jpg)
(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
![Page 49: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/49.jpg)
(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
![Page 50: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/50.jpg)
(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
![Page 51: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/51.jpg)
(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
![Page 52: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/52.jpg)
(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
![Page 53: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/53.jpg)
(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
![Page 54: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/54.jpg)
(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
![Page 55: X2 t05 03 parallel crosssections (2012)](https://reader034.vdocuments.us/reader034/viewer/2022042715/55869cbfd8b42a70728b45d8/html5/thumbnails/55.jpg)
(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