methods of volumes (1 Viewer)

[hatty]

hey

how do u rotate some curve around a line that isn't horizontal or vertical

eg. y = x

cheers

 

[Affinity]

same drill, slice the thing up, and find the volume of each slice..
and integrate.. I guess that's like treating the line y=x as the.. say u axis

 

[hatty]

but which direction are the slices?
horizontal or vertical?

 

[Affinity]

perpendicular to the line

 

[wogboy]

Best way to do such a thing is by slicing as normal, but instead of taking slices perpendicular to the x-axis, you will need to take slices perpendicular to y=x. Parametrising the curve you want to rotate is the way to go as you'll see:

e.g. to find the volume of rotating y=x^2 (0 < x < 1) around y=x:
y = x^2 may be parametrised as: (t,t^2) (i.e. any point that lies on y=x^2 can be represented as (t,t^2) ).
The distance between (t,t^2) and the line x - y = 0 is d = |t - t^2|/sqrt(2) (from the 2U distance formula), so:
volume = I {0 -> 1} pi * (t - t^2)^2 / 2 dt (note that the limits of integration are 0 to 1 since your integrating each slice from t=0 up to t=1). You can solve the integral yourself.

Challenge: See if you can find the volume of y=x (0 < x <1) rotated around y = x^2. Don't worry about solving the integral if it is difficult.

Hint: You should end up with some banana like shape.

 

[hatty]

wow
thanks alot mate