Answer
$2.4 \times 10^{18} \ kilograms $
Work Step by Step
The spherical coordinates system can be expresses as:
$x=\rho \sin \phi \cos \theta $ and $ y=\rho \sin \phi \sin \theta ; z=\rho \cos \phi$ and $\rho=\sqrt {x^2+y^2+z^2}$
or, $\rho^2=x^2+y^2+z^2$
Now,
$Mass; M= \int_0^{\pi} \int_0^{2 \pi} \int_{6.370 \times 10^{6}}^{6.375 \times 10^{6}} (619.09 -0.00007 \rho) \rho^2 \sin \phi d\rho \ d\theta d \phi$
or, $=\int_0^{\pi} \sin \phi d \phi
\int_0^{2 \pi} d\theta \int_{6.370 \times 10^{6}}^{6.375 \times 10^{6}} (619.09 \rho^2 -0.00007 \rho^3 \ d\rho$
or, $= 4 \pi [206.36 \rho^3 -0.00002425 \rho^4 ]_{6.370 \times 10^{6}}^{6.375 \times 10^{6}} $
By using a calculator, we have:
$Mass; M =4 \pi [206.36 \rho^3 -0.00002425 \rho^4 ]_{6.370 \times 10^{6}}^{6.375 \times 10^{6}}\\ \approx 2.4 \times 10^{18} \ kilograms $
