Derive-a-formula-of-volume-of-right-circular-cone-when-the-formula-of-volume-of-cyllinder-is-given-

Question Number 3714 by Rasheed Soomro last updated on 19/Dec/15

D erive a formula of volume of right circular cone

when the formula of volume of cyllinder is given .

Answered by Filup last updated on 19/Dec/15

I^{'} m not sure how to answer the question

given . But I can show it this way :

Line y = m x

h e i g h t = r

b a s e l e n g t h = h

(i t s a s i d e w a y s c o n e a g a i n s t t h e x a x i s)

∴ m = \frac{r}{h}

y = \frac{r}{h} x

V = π \int_{a}^{b} y^{2} d x

∴ V = π \int_{0}^{h} {(\frac{r}{h} x)}^{2} d x

V = π r^{2} \frac{1}{h^{2}} \int_{0}^{h} x^{2} d x

V = π r^{2} \frac{1}{h^{2}} (\frac{1}{3} h^{3} - 0)

∴ V = \frac{1}{3} π r^{2} h

Commented by RasheedSindhi last updated on 19/Dec/15

I n t e g r a l i s u s e d f o r a r e a .

H o w i s i t u s e d f o r v o l u m e ?

W h i c h l i n e i s y = m x ?

Commented by Filup last updated on 19/Dec/15

f o r y = f (x), volume is found by

rotating around either the x or y axis .

Rotate around x axis :

V = π \int_{a}^{b} y^{2} d x

y a x i s :

V = π \int_{a}^{b} x^{2} d y

a and b are values on the axis which

you are rotating around . S i m i l a r l y :

V = π \int_{x_{1}}^{x_{2}} y^{2} d x

V = π \int_{f (x_{1})}^{f (x_{2})} x^{2} d y

For example :

y = x^{2}

Rotating the curve around the y axis,

you have a shape that resembles a bowl .

Say the bowl is 4 units high .

y = x^{2} (r o t a t e a r o u n d y ∴ h a v e t o

t r a n s p o s e f o r x)

V = π \int_{a}^{b} x^{2} d y

y = x^{2}

x^{2} = y

V = π \int_{0}^{4} x^{2} d y

V = π \int_{0}^{4} y d y

V = \frac{1}{2} π 16

V = 8 π

Commented by Filup last updated on 19/Dec/15

Go onto the KhanAcademy website .

They have amazing tutorials and lessons

on calculating the volume of curves

rotated around an axis .

Just search it on google :)

Commented by RasheedSindhi last updated on 19/Dec/15

T h^{α} n k S!

Commented by Rasheed Soomro last updated on 22/Dec/15

A n o t h e r a p p r o a c h m a y b e a s g i v e n

b e l o w :

C o n s i d e r a c y l l e n d e r a n d a c o n e b o t h

e r e c t u p o n t h e i r c i r c u l a r b a s e s .

N o w a s w e g o a b o v e, t h e r a d i u s r e m a i n s

c o n s t a n t i n c y l l e n d e r b u t r a d i u s g o e s

t o d e c r e a s e m o r e a n d m o r e a n d u l t i m a t e l y

b e c o m e s z e r o a t i t s v e r t e x i n c a s e o f c o n e .

T h u s w e c a n s a y r = c i n c y l l e n d e r a n d

r \to 0 i n c o n e .

S o t h e v o l u m e o f c o n e i s a l i m i t o f c y l l e n d e r

v o l u m e a s r \to 0 ? ? ?