boitshupoO

2020-12-28

How are triple integrals defined in cylindrical and spherical coor-dinates? Why might one prefer working in one of these coordinate systems to working in rectangular coordinates?

Nicole Conner

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 $\left(r,\theta ,z\right)\in t\stackrel{^}{r}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\theta$ are polar coordinates for the vertical projection of P on the xy-plane with $r\ge \theta$ and z is the rectangular vertical coordinate. The equations related to the rectangular coordinates (x, y, z) and cylindrical coordinates $\left(r,\theta ,z\right)$ are, $x=r\mathrm{cos}\theta ,y=r\mathrm{sin}\theta ,z=z,{r}^{2}={x}^{2}+{y}^{2}$ and $\mathrm{tan}\theta =\frac{y}{x}$ Step 3 The spherical coordinates represent a point P in space by ordered triples $\left(p,\varphi ,\theta \right)$ in which, p is the distance from P to the origin $\left(p\ge 0\right)$ $\varphi$ is angle $over\to \left\{OP\right\}$ makes with the positive z-axis $\left(0\le \varphi \le \pi \right)$ $\theta$ is the angle from cylindrical coordinates. Step 4 The equations relating spherical coordinates to Cartesian and ctlindrical coordinates are, $r=p\mathrm{sin}\varphi$
$x=r\mathrm{cos}\theta =p\mathrm{sin}\varphi \mathrm{cos}\theta$
$z=p\mathrm{cos}\varphi$
$y=r\mathrm{sin}\theta =p\mathrm{sin}\varphi \mathrm{sin}\theta$
$p=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}=\sqrt{{r}^{2}+{z}^{2}}$
$\mathrm{tan}\theta =\frac{y}{x}$ 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.

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