IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

vinod

Find The VOLUME Of The solid formed by rotating

the curve around ox ( o is the centre) axis, the

curve is formed (bounded) by Y=X³ , X=0 , X=1.

Solve this......pls

The answer is

/2......i just wanna know the procedure...

vineet

Hi vinod i got a physics solution for a maths question,

find the y co-ordinate center of mass (COM) of the region of curve bounded by X=1 and X=0 ......................... u will get as (1/4).

Now ,
volume = 2

*(y co-ordinate) =

/ 2

Its bcos the curve is rotated about X axis and distance between X axis and COM is y co- ordinate.

If curve is rotated about Y axis V = 2

* (x co-ordinate)

vineet

Hey Vinod did u get that ?????????????????????

vinod

I THINK SO.............but can u pls give me the mathematical solution.....pls...

vinod

Pls any experts solve it.......

bestmust

the solution is very simple probably the imagination for you have been difficult ....

ok so now imagine according to me....

the figure is a compilation of many discs or probably cylinders with very small height (dx) above one other...

the radius of the disk (or cylinder) is y component and height is dx .....

so volume is (pi)(r^2)h = (pi)(y^2)dx

now to find the total volume we need to integrate the above function from 0 to 1 ... u might get the answer ....

kvenkan

the volume of a solid formed by rotating a curve around x-axis is

_[a]

^[b] y² dx

in this case it is

V=

_[0]

^[1] x⁶ dx

=

[ x⁷/ 7]₀¹ =

/7 cubic units

the ans is not pi/2

i'm pretty sure about it

pls rate me

zehra

using

V =

_{[ ]}

^{[ ]}

y^2 dx

lower limit 0 & upper limit 1. then putting y=x^3

Then we get,

V=

_{[ ]}

^{[ ]}

x^6 dx =

/7

Ans. /7

vinod

No, the ans is not

/7...i tooooooo had got the same ans when i solved, now let me say this question is objective type where the option given were :

/2 ,

/3,

/4,

/6

and the correct ans is

/2

itz give in the ansbook

but the detiled soln is not given there!!!!!

None of the experts r interested .....

!!!!

iitkgp_bipin

Consider a disc of thickness dx formed when the solid is sliced by a plane parallel to yz plane.

Its volume =

y²dx=

x⁶dx

Total volume = _[0]

^[1]

x⁶dx =

/7

The options which the book has provided may be wrong.

You can check solid of revolution to confirm the answer at :

http://en.wikipedia.org/wiki/Solid_of_revolution

avinash.sharma

iitkgp_bipin is right make it more clear by figures and description ( A general and perticular solution)

Figure 1 is the bounded area and figure 2 is the volume produced by rotating the area along x axis by 360 degree. Lets take a small cross section as in figure 3 of height h= dx having radius r=y then we can write the volume of such cylinder as

dV = p r²h = p y²dx

To calculate total volume we integrate this volume along dx when x moves from 0 to +1

V = ò p y²dx = ò p [f(x)]²dx where y =f(x) it is a general solution. In this integration you can put any y=f(x) and limits of x . In our case now put your curve y=x³ then

V = ò p [x³]²dx ( x moves 0 to +1 )

= ò p x⁶dx ( x moves 0 to +1 )

= p [x⁷/7] ( x moves 0 to +1 ) = p ( 1/7 ) = p/7

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