I am asked to find the mass of the cone. The cone has height 6, radius at the base 4 and the density function is $4+z$. I am asked to use cylindrical coordinates.
For this I reason I should solve the triple integral $$\int_0^6 \int_0^{2\pi} \int_0^4 (4+z)rdrd\theta dz $$ But, I know that this cannot be correct because the radius of the cone varies. How can I describe the radius of the cone?
[Math] Find the mass of a cone.
calculus
Best Answer
Assuming that the vertex of the cone is taken as the origin and that the axis of the cone is the $z$-axis, the cone can be described as:
$$x^2+ y^2 \le \frac{R^2}{h^2} z^2$$
Where $R$ is the radius of the base and $h$ is its height.
Then, in cylindrical coords, $r^2 \le \frac{R^2}{h^2}z^2$. I.e. $0 \le r \le \frac{R}{h} z$.
So, $z$ runs from $0$ to $6$, $r$ from $0$ to $(R/h)z$ and $\theta$ from $0$ to $2 \pi$.