Find the mass of the sphere in Example 3 if its density at a point is proportional to the distance from the point to the origin.
Question 5.7.4: Find the mass of the sphere in Example 3 if its density at a...
We have density =\alpha \sqrt{x^{2}+y^{2}+z^{2}}=\alpha \rho. Thus
\begin{aligned}\mu &=\int_{0}^{2 \pi} \int_{0}^{\pi} \int_{0}^{a} \alpha \rho\left(\rho^{2} \sin \varphi\right) d \rho d \varphi d \theta=\frac{\alpha a^{4}}{4} \int_{0}^{2 \pi} \int_{0}^{\pi} \sin \varphi d \varphi d \theta \\&=\frac{\alpha a^{4}}{4}(4 \pi)=\pi a^{4} \alpha\end{aligned}Related Answered Questions
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