# Parametric Volumes 1

https://isaacphysics.org/questions/param_vol_rev_1?board=907569dd-74c5-47f2-9f6c-64066dafc922&stage=a_level

I managed to get the answer to part e, but I am unsure if I used the right method?
I calculated y=2+sqrt(x/2) to get the "upper part" of the quadratic curve and integrated this for a full rotation about the x-axis.

I then did my answer subtract the answer from part b (so 272pi/3 - 16pi/3=256pi/3 which turned out to be correct)

Is there a quicker way or did I use the correct method?
Help much appreciated
(edited 1 month ago)
Original post by mosaurlodon
https://isaacphysics.org/questions/param_vol_rev_1?board=907569dd-74c5-47f2-9f6c-64066dafc922&stage=a_level
I managed to get the answer to part e, but I am unsure if I used the right method?
I calculated y=2+sqrt(x/2) to get the "upper part" of the quadratic curve and integrated this for a full rotation about the x-axis.
I then did my answer subtract the answer from part b (so 272pi/3 - 16pi/3=256pi/3 which turned out to be correct)
Is there a quicker way or did I use the correct method?
Help much appreciated

Id guess they wanted you to do the volumes parametrically so
Int pi y(t)^2 dx/dt dt
over the appropriate limits for t. Gives the same result and its not hugely simpler, but its probably more direct.
Oh I see... thanks

how would you geometrically interpret the volume using Int pi y(t)^2 dx/dt dt, since the upper bound is -2 > the lower bound of 0, do you just think of it by swapping the upper and lower bounds and multiplying this volume by negative one, since
Original post by mosaurlodon
Oh I see... thanks
how would you geometrically interpret the volume using Int pi y(t)^2 dx/dt dt, since the upper bound is -2 > the lower bound of 0, do you just think of it by swapping the upper and lower bounds and multiplying this volume by negative one, since

Just use the limits 0 to -2. You could reverse them and negate the integral, but it ends up the same. You can always parametrically define it in terms of -t and it would make "more sense" for this curve segment, but reality is youre simply flipping the limits from x to t and the values are fairly irrrelevant as long as theyre correct.
(edited 1 month ago)