Find the Mass

Calculus Level 5

$x = r\cos(\theta)\hspace{1cm}y = r\sin(\theta)\hspace{1cm}z = z$ $0\leq\,r\leq\sqrt{z}\hspace{1cm}0\leq\,\theta\leq2\pi\hspace{1cm}0\leq\,z\leq1$

A solid object exists within the standard $(x,y,z)$ coordinate system. Its geometry is parametrized as shown above.

The object has a mass density $\large {\rho = e^{z^{2}}}$ , where $e$ is Euler's number . The objects mass can be expressed as:

${\dfrac{\pi}{A} (e - B)},$

where $A$ and $B$ are integers , determine $A+B$ .

2 solutions

First Last
Sep 3, 2016

Using cylindrical coordinates, with the substitutions provided,

$\displaystyle x = r\cos\theta \quad y = r\sin\theta \quad z = z$

we set up a triple integral with the mass function as the integrand. The Jacobian for this variable change is just $r$ .

mass $\displaystyle = \int_{0}^{1}\int_{0}^{\sqrt{z}}\int_{0}^{2\pi}e^{z^2}r \quad d\theta dr dz = \boxed{\frac{\pi (e - 1)}{2}}$

Steven Chase
Sep 2, 2016