Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can someone help me with the following one?

I need to calculate the mass of the surface $x+y+z=a $ , $a>0$ intersected with the cylinder $x^2 + y^2 = R^2 $ where the density is $ u(x,y,z)= (x^2+y^2)^2 + z $ .

As far as I know, I need to calculate : $ \iint_S u(x(u,v),y(u,v),z(u,v)) \,du\,dv $ where $(x(u,v), y(u,v), z(u,v))$ is the parameterization of our surface .

Can someone help me find the parameterization of such a surface?

Please help me. I'm really frustrated. Thanks !

share|cite|improve this question
The intersection is not a surface, it's a curve (an ellipse, actually). So what is you density function $u$? It looks to me that if it's an area density, the curve will have mass $0$; if it's a mass attached to each point, the cylinder has infinite mass… – jathd Jan 17 '13 at 8:40
what if it is a mistake and we are actually deaing with $x^2 + y^2 \leq R^2 $ ? I think this is what they meant (although they didn't write it... ). Can you help me in this particular meaning? Thanks ! – theMissingIngredient Jan 17 '13 at 8:44
up vote 2 down vote accepted

Assuming the surface of interest is the intersection between the place $x+y+z=a$ and the "full" cylinder $x^2+y^2\leqslant R^2$, here's a parametrization of the surface. You can describe it as $$ S = \{(x,y,z)\in\mathbf R^3 : x^2+y^2\leqslant R^2,~z=a-x-y\}, $$ so you can use cylindrical coordinates: $x=r\cos(\theta)$, $y=r\sin(\theta)$, $z=a-x-y=a-r\cos(\theta)-r\sin(\theta)$ for $r\in[0,R]$ and $\theta\in[0,2\pi]$.

So the mass you're looking for is $$ M = \int_{r=0}^R\int_{\theta=0}^{2\pi}(r^4+a-r\cos\theta-r\sin\theta)\,\mathrm dr\,\mathrm d\theta, $$ which shouldn't be too hard to compute.

share|cite|improve this answer
Thanks a lot ! !!!!!!!!!!!!!!!!!!!! – theMissingIngredient Jan 17 '13 at 10:45

Parametrizing your surface is easy: it's a plane. So you just put

$$ \varphi (x,y) = (x, y, a -x-y) $$

and the limits of integration are those of the circle

$$ x^2 + y^2 \leq R^2 \ . $$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.