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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)
Reply 1
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
Reply 3
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)

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