Expert: Paul Klarreich - 6/17/2008

Question
Sorry didn't mean to put it on private before, i have two questions about solid of revolution, from a practice exam before my final exam. They are:

Find the volume of the solid of revolution obtained by rotating, a full turn about the x-axis, the area between the x-axis and the curve y = 2 sin x, for x
∈ [π/4 , 3π/4 ].

and:

S is a solid of revolution in 3-dimensions, formed by rotating a full turn about the y-axis, the region in the first quadrant of the (x, y)-plane bounded by the interval [1, 2] on the y-axis, and the curve x = (2 − y)(y − 1)^2  

Need to find the volume, moment My and centre of mass of the solid object S .  

Answer
Questioner:   James
Category:  Advanced Math
Private:  No
 
Subject:  Solid of revolution questions
Question:  Sorry didn't mean to put it on private before, i have two questions about solid of revolution, from a practice exam before my final exam. They are:

Find the volume of the solid of revolution obtained by rotating, a full turn about the x-axis, the area between the x-axis and the curve y = 2 sin x, for x
∈ [π/4 , 3π/4 ].

and:

S is a solid of revolution in 3-dimensions, formed by rotating a full turn about the y-axis, the region in the first quadrant of the (x, y)-plane bounded by the interval [1, 2] on the y-axis, and the curve x = (2 − y)(y − 1)^2  

Need to find the volume, moment My and centre of mass of the solid object S .
...............................................
Hi, James,

For this one:

Find the volume of the solid of revolution obtained by rotating, a full turn about the x-axis, the area between the x-axis and the curve y = 2 sin x, for x
∈ [π/4 , 3π/4 ].

[You have to write it this way:]
pi/4 <= x <= 3pi/4

I have trouble making those special symbols and getting them through the site.  

Use disks.  Your 'typical' disk has:

dV = pi r^2 h, where

radius = 2 sin x
h = thickness = dx.

Integrate:

{3pi/4
|       4 sin^2 x dx
}pi/4

Use the half-angle trick:

{3pi/4   1 - cos 2x
|      4 ---------- dx
}pi/4        2


{3pi/4  
|     (2 - 2 cos 2x) dx
}pi/4      


2x - sin 2x, from pi/4 to 3pi/4

= (3pi/2 - sin(3pi/2)) - (pi/2 - sin(pi/2))

= (3pi/2 - (-1)) - (pi/2 - (1))

= 3pi/2 + 1 - pi/2 + 1

= pi + 2 = 5.14, about.

.............................................


S is a solid of revolution in 3-dimensions, formed by rotating a full turn about the
y-axis, the region in the first quadrant of the (x, y)-plane bounded by the interval [1, 2] on the y-axis, and the curve x = (2 − y)(y − 1)^2  

........
I think you take as your 'sample' a slice that goes from x = 0 to x = (2 − y)(y − 1)^2,
and let y go from  1 to 2.  [See the sample -- I drew  y = (2 - x)(x - 1)^2, and formed its inverse -- the dotted graph.]

Need to find the volume, moment My and centre of mass of the solid object S .

For the volume, use disks again:

r = x = (2 − y)(y − 1)^2
h = thickness = dy.

Integrate:
{2
|   (2 − y)(y − 1)^2 dy = V
}1

That should be routine.  Messy, perhaps, but all you have to do is multiply out and integrate the polynomial.

For the moment in the y-direction, you multiply by y:

{2
|  y (2 − y)(y − 1)^2 dy = My
}1

[You can do that, too.]

For the centroid, you are supposed to get  (x-bar, y-bar).  x-bar is obviously zero, and y-bar is just  My/V.

That should do it.