I am asked the following question:
Find the volume of sphere $x^2+y^2+z^2 \leq a^2$ contained in $z \geq 0$, $y \leq x$ and $y \geq 0$
What i gather from the information presented is:
$$ \begin{align*} z \geq 0 &\Rightarrow 0 \leq \phi \leq \pi/2\\ y \geq 0 \text{ and } y \leq x &\Rightarrow 0 \leq \theta \leq \pi/4\\ x^2+y^2+z^2 \leq a^2 &\Rightarrow 0 \leq \rho \leq a \end{align*} $$
Ist that correct? Should I evaluate the following?
$$ \begin{align*} \int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{a} \rho^2 \sin(\phi) \ d\rho d\phi d\theta \end{align*} $$
Thank you.
From a purely geometric point of view, is it clear that you should be getting one $\;16\,-$ th of the total volume of the sphere?:
$$\int_{0}^{\pi/4} \int_{0}^{\pi/2} \int_{0}^{a} \rho^2 \sin(\phi) \ d\rho d\phi d\theta=\frac\pi4\int_0^{\pi/2}\sin\phi\left.\frac13\rho^3\right|_0^a=$$
$$=\left.\frac{a^3\pi}{12}(-\cos\phi)\right|_0^{\pi/2}=\frac{a^3\pi}{12}=\frac1{16}\cdot\color{red}{\frac43a^3\pi}$$
as expected.