Question

Find the mass of the solid cylinder Dequals​{(r,theta​,z): 0less than or equalsrless than or equals2​, 0less than or equalszless than or equals10​} with density rho​(r,theta​,z)equals1plusStartFraction z Over 2 EndFraction . Set up the triple integral using cylindrical coordinates that should be used to find the mass of the solid cylinder as efficiently as possible. Use increasing limits of integration. Integral from 0 to nothing Integral from nothing to nothing Integral from nothing to nothing (nothing )font size decreased by 3 dz font size decreased by 3 dr font size decreased by 3 d theta

Answer 1

The mass of the cylinder D is obtained by integrating the density function over D:

$\displaystyle\iiint_D\rho(r,\theta\,z)\,\mathrm dV$

With $\rho(r,\theta,z)=1+\frac z2$, and

$D=\left\{(r,\theta,z)\mid 0\le r\le2,0\le z\le10\right\}$

the mass would be

$\displaystyle\int_0^{2\pi}\int_0^2\int_0^{10}1+\frac z2\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=4\pi\int_0^{10}1+\frac z2\,\mathrm dz$

$4\pi\left(z+\dfrac{z^2}4\right)\bigg|_0^{10}$

$=4\pi\left(10+\dfrac{100}4\right)=\boxed{140\pi}$