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I need to express the following regions in both cylindrical and spherical coordinates. I am not sure what to do here exactly, in (a) for example, should I substitute the Cartesian equation of the sphere and the given equation $z$ with the cylindrical and spherical coordinates, but then what?

(a) The volume that lies inside the sphere R (centred on the origin) and is also above the conical surface $z=\sqrt{x^2+y^2}$ .

(b) A circular cone of base radius $R$ and height $h$. Take the cone to be oriented with the vertex at the origin and the base at height $h$ above (i.e. at $z=h$ in Cartesian coordinates).

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(a) would be expressed in Cartesian coordinates as $x^2+y^2+z^2<R^2 \land z>\sqrt{x^2+y^2}$. All you need to do is insert the coordinate transformation in these formulas, e.g. for cylindrical coordinates $x=r\cos(\theta),y=r\sin(\theta),z=z$ gives $r^2+z^2<R^2 \land z>r$

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  • $\begingroup$ Thanks. Cylindrical coordinates are ($r,\theta,z$ ) and I usually specify the coordinates as inequalities. For example I know that $z>r$, $r<R-z$, but what is $\theta$, is it from $0$ to $2pi$ ? $\endgroup$ – NeoXx Mar 2 '16 at 1:18

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