Calculate the volume enclosed from above by a sphere of radius $\\r = R$ and from below by the plane $\\z = \frac{R}{2}$:
a) Using cylindrical coordinates.
b) Using spherical coordinates.
c) Using cartesian coordinates.
I am struggling to find the correct limits of integration. This is what I have been able to come up with so far :
a) $\phi:0\to 2\pi$
$ \\ z:\frac{R}{2}\to R$
$\rho$:?
b) I do not know where to start
c) $\\z:\frac{R}{2} \to R $
I am unsure about $x,y$
a) Cylindrical coordinates
$$\int_0^{2\pi}d\phi\int_0^{\frac{\sqrt3}{2}R}\rho d\rho\int_{\frac R2}^{\sqrt{R^2-\rho^2}}dz$$
b) Spherical coordinates
$$\int_0^{2\pi}d\phi\int_0^{\frac\pi3}\sin\theta d\theta \int_{\frac R{2\cos\theta}}^{R}r^2dr$$
c) Cartesian coordinates
$$\int_{-\frac{\sqrt3}{2}R}^{\frac{\sqrt3}{2}R}dx\int_{-\sqrt{ \frac{3}{4}R^2 -x^2}}^{\sqrt{ \frac{3}{4}R^2 -x^2}}dy \int_{\frac R2}^{\sqrt{ R^2 -x^2-y^2}}dz$$