the mother of all p2 volumes of rev. question

Find the volume when the following shaded region is rotated 360 degrees about the y-axis.


Book says answer= pi.


well i worked out the vol of shaded region on one side of the y axis- do i need to double this or is this the final answer?? because the shaded is symetrical about yaxis, and surely the same volume is achieved by not doubling??

omfg!! i would guess that it should equal half of volume... but how do you calculate it in the first place.???? Could you give a step-by-step please...!!

Find the volume when the following shaded region is rotated 360 degrees about the y-axis.


Book says answer= pi.


well i worked out the vol of shaded region on one side of the y axis- do i need to double this or is this the final answer?? because the shaded is symetrical about yaxis, and surely the same volume is achieved by not doubling??

volume=pi[INT(2-y) (between 1 and 2)+INTy (between 0 and 1)]
=pi((2y-y^2/2)+(y^2/2))
=pi(4-2-2+1/2+1/2)
=pi!!!

so you dont need to double?

I got it!!!!!!!! vol = pi*int(2-x^2)^2 = 19/3pi - 10/3pi = 3pi (limits 1 and -1)

other volume = pi*int(x^2)^2 = pi (limits are 0 and -1). Do this twice as it is a reflection (limits 1 and 0). Add these together to get 2pi. (if done with limits 1 and -1, they will cancel each other out to give 0)

so volume shaded = 3pi-2pi
= pi

volume=pi[INT(2-y) (between 1 and 2)+INTy (between 0 and 1)]
=pi((2y-y^2/2)+(y^2/2))
=pi(4-2-2+1/2+1/2)
=pi!!!

shouldnt you square f(x) when finding the volume of revolution?

he did, but didnt write all the steps

y = x^2
y = 2 - x^2
x^2 = 2 - x^2
2x^2 = 2
x = +/- 1

Area = (1/-1){ (2 - x^2) dx - (1/1){ x^2 dx
= [2x - x^3/3](1/-1) - [x^3/3](1/-1)
= {[2*1 - 1^3/3] - [2*-1 - -1^3/3]} - {[1^3/3] - [-1^3/3]}
= {[10/6] - [2/3]}
= 1 square unit.

Square and multiply by pi:

pi.1^2 = pi cube units.
Great question!

This is driving me crazy... I'm trying to work through yours to see where I've gone wrong, but this little bit is driving me insane...

= {[2*1 - 1^3/3] - [2*-1 - -1^3/3]} - {[1^3/3] - [-1^3/3]}
= {[10/6] - [2/3]}

How the hell do you get 10/6 ? I've tried it over and over again, I get 10/3!!!

(2 - 1/3) - (-2 + 1/3) = 10/3
aaarrgh! whyyyyyyy?

I can't see myself... I guess it's a mistake.