Find the volume of the solid bounded by the surfaces: $x² + y² + z² = a²$ and $x² + y² \geq a|x|$, ($a > 0$)

Question

Find the volume of the solid bounded by the surfaces: $x^2 + y^2 + z^2 = a^2$ and $x^2 + y^2 \geq a|x|, (a > 0)$

Answer 1

Hint: $$V = \int_{0}^a\int_{0}^{\pi}\int_{-\pi}^{\pi} \mathbf 1_{r^2\sin^2 \theta \ge a r\sin\theta\left|\cos\phi\right|} \mathrm d\varphi \,\sin\theta\mathrm d\theta\, r^2\mathrm dr$$