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