Questions: Integrals for Mass Calculations A spherical solid, centered at the origin, has radius 81 and mass density δ(x, y, z)=87-(x^2+y^2+z^2). Find its mass. ∫ ∫ ∫ δ(x, y, z) dV=∫ ∫ ∫ d rho d theta d phi= For your answers theta= theta, rho= rho, phi= phi.

Solution Steps

To find the mass of the spherical solid, we need to integrate the mass density function over the volume of the sphere. We will use spherical coordinates for this integration. The mass density function is given as \(\delta(x, y, z) = 87 - (x^2 + y^2 + z^2)\). In spherical coordinates, \(x^2 + y^2 + z^2 = \rho^2\), where \(\rho\) is the radial distance from the origin. The limits for \(\rho\) will be from 0 to 81, for \(\theta\) from 0 to \(2\pi\), and for \(\phi\) from 0 to \(\pi\). The volume element in spherical coordinates is \(dV = \rho^2 \sin(\phi) d\rho d\theta d\phi\).

The mass density function for the spherical solid is given by

\[ \delta(x, y, z) = 87 - (x^2 + y^2 + z^2). \]

In spherical coordinates, this becomes

\[ \delta(\rho) = 87 - \rho^2. \]

In spherical coordinates, the volume element is expressed as

\[ dV = \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta. \]

The limits for the integration are as follows:

• For \(\rho\): from \(0\) to \(81\),
• For \(\phi\): from \(0\) to \(\pi\),
• For \(\theta\): from \(0\) to \(2\pi\).

The mass \(M\) of the spherical solid is calculated using the integral:

\[ M = \iiint \delta(\rho) \, dV = \int_0^{2\pi} \int_0^{\pi} \int_0^{81} (87 - \rho^2) \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta. \]

Upon evaluating the integral, we find:

\[ M = -\frac{13638901824\pi}{5}. \]

The numerical value of the mass is

\[ M \approx -8569574754.6622. \]

Final Answer