- #1
HJ Farnsworth
- 128
- 1
Greetings,
If we have a function y=f(x), we can calculate the surface area traced by that function when rotating about the x-axis as
1: [itex]S=∫dx\,2πf(x)[/itex],
which makes perfect sense to me. I am told that, if we have x=x(t) and y=y(t), the equivalent expression is
2: [itex]S=∫dt\,2πf(x)\sqrt{(dx/dt)^2+(dy/dt)^2}[/itex].
I find (2) a bit suspicious, since it seems that we are now integrating along the parametrized curve itself, rather than along the x-axis. In other words, it seems to me that (2) is equivalent to
3: [itex]S=∫dl\,2πf(x)[/itex],
which is not the same as (1).
Furthermore, in the case that our parametrization satisfies y=f(x), (2) becomes
4: [itex]S=∫dx\,2πf(x)\sqrt{1+(dy/dx)^2}[/itex],
which seems to me to brazenly contradict (1). In both (1) and (4), we are looking for the surface area traced out by rotating a curve y=f(x) about the x-axis, and yet we have two different expressions for the area.
What am I missing here? More to the point, what is it that makes (1) and (4) not contradict each other?
Thanks for the help!
-HJ Farnsworth
If we have a function y=f(x), we can calculate the surface area traced by that function when rotating about the x-axis as
1: [itex]S=∫dx\,2πf(x)[/itex],
which makes perfect sense to me. I am told that, if we have x=x(t) and y=y(t), the equivalent expression is
2: [itex]S=∫dt\,2πf(x)\sqrt{(dx/dt)^2+(dy/dt)^2}[/itex].
I find (2) a bit suspicious, since it seems that we are now integrating along the parametrized curve itself, rather than along the x-axis. In other words, it seems to me that (2) is equivalent to
3: [itex]S=∫dl\,2πf(x)[/itex],
which is not the same as (1).
Furthermore, in the case that our parametrization satisfies y=f(x), (2) becomes
4: [itex]S=∫dx\,2πf(x)\sqrt{1+(dy/dx)^2}[/itex],
which seems to me to brazenly contradict (1). In both (1) and (4), we are looking for the surface area traced out by rotating a curve y=f(x) about the x-axis, and yet we have two different expressions for the area.
What am I missing here? More to the point, what is it that makes (1) and (4) not contradict each other?
Thanks for the help!
-HJ Farnsworth