Volume of Object

1. Nov 8, 2006

Mindscrape

The top of an object is bounded by $$z=4-4(x^2+y^2)$$ and the bottom is bounded by $$z=(x^2+y^2)^2 -1$$, find the volume.

So I converted to polar coordinates such that
top: $$z=4-4r^2$$
bot: $$z=r^2 - 1$$

Then I integrated the first quadrant and multiplied by 4, with the bounds of dzdrdø such that the volume was:
$$V= 4[ \int_{\theta=0}^{\pi/2} \int_{r=0}^{r=1} \int_{z=0}^{z=4-4r^2} r dz dr d \theta + \int_{\theta=0}^{\pi/2} \int_{r=0}^{r=1} \int_{z=r^2-1}^{z=0} r dz dr d\theta ]$$

The answer I got was 7pi/2, but the answer from the book is 8pi/3. I think I have done all the arithematic right, but did I set it up right?

Last edited: Nov 8, 2006
2. Nov 8, 2006

quasar987

what is the region of integration in terms of x&y?

3. Nov 8, 2006

Mindscrape

It isn't given, the problem just gave the object in terms of its top and bottom. I sketched it out, and the object is essentially one of those plastic easter eggs (where the top is longer and narrower than the bottom).

But, just going back then in terms of x&y the integration will be the very sticky:
$$V= 4[ \int_{y=0}^{y=1} \int_{x=0}^{x=\sqrt{1-y^2}} \int_{z=0}^{z=4-4(x^2+y^2)} dz dx dy + \int_{y=0}^{y=1} \int_{x=0}^{x=\sqrt{1-y^2}} \int_{z=(x^2+y^2)^2-1}^{z=0}dz dx dy]$$
Which does not look appealing to calculate.

Last edited: Nov 8, 2006
4. Nov 8, 2006

quasar987

Ah, but $$z=(x^2+y^2)^2 -1$$ transforms into $$z=r^4 -1$$! power 4 not 2.

5. Nov 8, 2006

Mindscrape

Oh no, you are right! Oops.

Thanks.