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