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.