find an equation for surface of revolution

• Jun 5th 2012, 08:36 PM
icelated
find an equation for surface of revolution
I have a problem where i need to
find the equation for surface of revolution if the generating curve y = 2x + 1, is revolved about the y axis

The book gives this formula for revolving around the y axis

\$\displaystyle x^2 + z^2 = [ r(y)]^2\$

So, wouldnt i just plug y in and square it?
most of the problems have been done this way

Problem is i dont get the right answer.

\$\displaystyle 4x^2 -y^2 + 4z^2 + 2y - 1 =0\$

What do i need to do?
Thank you
• Jun 6th 2012, 07:35 AM
BobP
Re: find an equation for surface of revolution
The equation \$\displaystyle x^{2}+z^{2}=[r(y)]^{2}\$ is that of a pair of cones (or is it a single cone ?) with a common vertex and an axis of symetry along the \$\displaystyle y\$ axis. The cones open away from each other from the common vertex. The \$\displaystyle r\$ in the equation is the radius of the cone and the way that it is written \$\displaystyle r(y)\$ says that it is a function of \$\displaystyle y.\$ The radius will be zero when \$\displaystyle y=1\$ and \$\displaystyle 1/2\$ when \$\displaystyle y=0,\$ (or \$\displaystyle y=2),\$ these coming from the generating line \$\displaystyle y=2x+1.\$
What you are looking for then is a relationship of the form \$\displaystyle r=ay+b\$ for some values of \$\displaystyle a\$ and \$\displaystyle b\$ such that \$\displaystyle r(0)=1/2\$ and \$\displaystyle r(1)=0.\$
That turns out to be \$\displaystyle r=(1-y)/2\$ and when you substitute that into the given equation it simplifies to your given answer.
• Jun 6th 2012, 10:23 AM
HallsofIvy
Re: find an equation for surface of revolution
BobP, mathematically, its a single cone. Each part, that in "standard English" is called a cone, is a "nappe" of that cone.

icelated, the "r(y)" is the function x= f(y) in the plane. Since you are given the line y= 2x+ 1, x= r(y)= (y- 1)/2. That is what should be squared, \$\displaystyle x^2+ z^2= ((y- 1)/2)^2= (y^2- 2y+ 1)/4\$ so that \$\displaystyle 4x^2- y^2+ 4z^2- 2y= 1\$.
• Jun 6th 2012, 11:45 AM
icelated
Re: find an equation for surface of revolution
@HallsofIvy thank you that makes perfect sense.