Cylindrical and spherical coordinates

In cylindrical coordinates, \(r\) is the distance to the \(z\)-axis, \(\theta\) the angle of the projection onto the \(xy\)-plane, and \(z\) the height, so \(r^2=x^2+y^2\) and \(\tan\theta=y/x\). They are polar coordinates plus a height: \(r=c\) is a cylinder, \(\theta=c\) a half-plane and \(z=c\) a horizontal plane. In spherical coordinates, \(\rho\) is the distance to the origin and \(\varphi\in[0,\pi]\) the angle from the positive \(z\)-axis; then \(\rho=c\) is a sphere, \(\varphi=c\) a cone and \(\theta=c\) the same half-plane. Also, \(\rho^2=x^2+y^2+z^2\) and \(r=\rho\sin\varphi\).

When integrating, each system brings its own volume factor: \(dV=r\,dz\,dr\,d\theta\) in cylindrical and \(dV=\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta\) in spherical coordinates. The factor is the volume of the small curved cell: in spherical coordinates its sides measure \(d\rho\), \(\rho\,d\varphi\) and \(\rho\sin\varphi\,d\theta\). The integral becomes \(\iiint_E f\,dV=\iiint f(\ldots)\,\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta\), with \(f\) rewritten in the new variables.

Guiding example: the volume of the ball of radius \(a\) is \(\int_0^{2\pi}\!\int_0^\pi\!\int_0^a\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta=\tfrac43\pi a^3\). Rule of thumb: use cylindrical coordinates if the solid revolves around the \(z\)-axis (cylinders, paraboloids \(z=r^2\)), and spherical ones if there are spheres or cones centered at the origin.

Typical mistakes: forgetting the factor \(r\) or \(\rho^2\sin\varphi\), swapping \(\theta\) and \(\varphi\) (other books name them the other way round, so fix the convention first), and letting \(\varphi\) run from \(0\) to \(2\pi\) instead of \(0\) to \(\pi\).