[Math] Finding cylinder mass using triple integrals

calculus

I'm trying to find mass of a cylinder using triple integrals.

The data I have:

  • height = $h$
  • base is a circle and its radius = $a$
  • the density in a point P is directly propotional to the distance of its basis.

I think that I can use $p(z) = k*z$

and that the basis formula is

$x^2 + y^2 = a^2$ (the circunference formula).

I also think that the inferior limit in $z$ integral is 0 and the superior limit is $h$.

But I can't figure out the x and y inferior and superior limits.

Someone can help me?

Best Answer

Ok, if you must use cartesian coordinates then you need to realise your cylinder is defined over $-a \leq x \leq a, -\sqrt{a^2 - x^2} \leq y \leq \sqrt{a^2 - x^2}, 0 \leq z \leq h$.

So your integral to calculate the mass of the cylinder will look like

\begin{equation} \textrm{mass} = \int_{x = -a}^{x = a} \int_{y = - \sqrt{a^2 - x^2}}^{y = \sqrt{a^2 - x^2}} \int_{z = 0}^{z = h} p(z)\ dzdydx. \end{equation}

Hope this helps.

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