Volumes Help (1 Viewer)

[Aysce]

This is one of the more easier volumes questions (cylindrical shells) but I'm still confused about what I have to find and also the dimensions of the strip or whatever..

So here it is:

Q. By taking strips parallel to the axis of rotation, use the method of cylindrical shells to find the volume of the solid obtained by rotating the region enclosed between the curve y= 4-x^2 and the lines x=2 and y=4 through one complete revolution about the x-axis.

I don't really want full working but maybe just a few "hints" otherwise it contradicts the whole point of doing this..

Thank you to all.

 

[math man]

well when you rotate it you will form an annulus cylinder, cylinder with hole in the middle.
Now the height of the cylinder will be 2-x and the radius of the cylinder is y with thickness
delta y. Now to work out the volume of that strip what you do is break that annulus cylinder
into a rectangular prism by slicing the annulus cylinder down one side. Now the length of the
rectangular prism will be the circumference of the top circle of the annulus cylinder which is
2(pie)y, hence the volume of the strip is deltaV=2(pie)(2-x)y(delta y). Im pretty sure you
can finish it from here.

 

[Aysce]

well when you rotate it you will form an annulus cylinder, cylinder with hole in the middle.
Now the height of the cylinder will be 2-x and the radius of the cylinder is y with thickness
delta y. Now to work out the volume of that strip what you do is break that annulus cylinder
into a rectangular prism by slicing the annulus cylinder down one side. Now the length of the
rectangular prism will be the circumference of the top circle of the annulus cylinder which is
2(pie)y, hence the volume of the strip is deltaV=2(pie)(2-x)y(delta y). Im pretty sure you
can finish it from here.

Can I just ask why the height of the cylinder is 2-x?

 

[math man]

If you draw your diagram right then the
region we are rotating is the one bounded
by y=4 and x=2 and so the distance along
the x direction will be the height when we
rotate that region about the x axis. Now
the length of the strip horizontally in that
region is 2-x which is the height, if you still
Don't understand I'll draw a diagram later when
I get home.

 

[Carrotsticks]

Can I just ask why the height of the cylinder is 2-x?

 

[math man]

Carrot pretty diagram but it shows wrong
region, question asked between y=4 and
x=2

 

[Aysce]

Carrot pretty diagram but it shows wrong
region, question asked between y=4 and
x=2

Oh i get it now but when I reach the final part of my working:

V= 2(pi) * Integral of y(2 - (4-y)^1/2) from 0 to 4, how am I supposed to integrate with y(4-y)^1/2 ?

 

[nightweaver066]

Oh i get it now but when I reach the final part of my working:

V= 2(pi) * Integral of y(2 - (4-y)^1/2) from 0 to 4, how am I supposed to integrate with y(4-y)^1/2 ?

Let u = 4 - y, etc.

 

[Aysce]

Let u = 4 - y, etc.

Damn.. forgot about that.. Hating volumes so much..

 

[Carrotsticks]

Carrot pretty diagram but it shows wrong
region, question asked between y=4 and
x=2

Oh woops lol didn't read properly.