How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working...

Step 1 Give the notes about how the triple integrals defined in cylindrical and spherical coordinates. Step 2 The cylindrical coordinates denotes a point P in space by ordered triples ${\displaystyle (r,\theta ,z)\in t\hat{r}\text{and}\theta }$ are polar coordinates for the vertical projection of P on the xy-plane with ${\displaystyle r\ge \theta }$ and z is the rectangular vertical coordinate. The equations related to the rectangular coordinates (x, y, z) and cylindrical coordinates ${\displaystyle (r,\theta ,z)}$ are, ${\displaystyle x=r\cos \theta ,y=r\sin \theta ,z=z,r^2=x^2+y^2}$ and ${\displaystyle \tan \theta =\frac{y}{x}}$ Step 3 The spherical coordinates represent a point P in space by ordered triples ${\displaystyle (p,\varphi ,\theta )}$ in which, p is the distance from P to the origin $(p\ge 0)$ ${\displaystyle \varphi }$ is angle ${\displaystyle over\to \left\{OP\right\}}$ makes with the positive z-axis ${\displaystyle (0\le \varphi \le \pi )}$ ${\displaystyle \theta }$ is the angle from cylindrical coordinates. Step 4 The equations relating spherical coordinates to Cartesian and ctlindrical coordinates are, ${\displaystyle r=p\sin \varphi }$
${\displaystyle x=r\cos \theta =p\sin \varphi \cos \theta }$
${\displaystyle z=p\cos \varphi }$
${\displaystyle y=r\sin \theta =p\sin \varphi \sin \theta }$
${\displaystyle p=\sqrt{x^2+y^2+z^2}=\sqrt{r^2+z^2}}$
${\displaystyle \tan \theta =\frac{y}{x}}$ Cylindrical coordinates are good for describing cylinders whose axes run along the z-axis and planes that either contain the z-axis or lie perpendicular to the z-axis. Surfaces like these have equations of constant constant coordinate value.