91Ó°ÊÓ

The density at every point is proportional to the square of the point’s distance from the $xy$-plane.

The region bounded above by the plane is given by equation$z=x$ and bounded below by the paraboloid by equation$z=x^2+y^2$.

Relation between rectangular and cylindrical coordinates is given by

$r=\sqrt{x^2+y^2},\tan \mathrm{θ}=\frac{y}{x},z=z$

And

$x=r\cos \mathrm{θ},y=r\sin \mathrm{θ},z=z$

Rectangular coordinates are $z=x$and$z=x^2+y^2$

Cylindrical coordinates are $z=r\cos \mathrm{θ}$and $z=r^2$

Cartesian limits are $x^2+y^2≤z≤x$

$x=r\cos \mathrm{θ}⇒z=x⇒z=r\cos \mathrm{θ}$

$r^2=x^2+y^2⇒z=x^2+y^2⇒z=r^2$

$⇒r^2≤z≤r\cos \mathrm{θ}$

$x^2+y^2=x\text{gives}{r}^2=r\cos \mathrm{θ}⇒r=0,r=\cos \mathrm{θ}→0≤r≤\cos \mathrm{θ}$

Also

$r=0\text{gives}\cos \mathrm{θ}=0⇒-\mathrm{π}/2≤\mathrm{θ}≤\mathrm{π}/2$

Cylindrical limits are

role="math" localid="1652385337856" $r^2≤z≤r\cos \mathrm{θ},0≤r≤\cos \mathrm{θ},-\mathrm{π}/2≤\mathrm{θ}≤\mathrm{π}/2$

The density at every point is proportional to the square of the point’s distance from the $xy$plane.

$⇒\mathrm{ÒÏ}=k{z}^2$

Required mass is $m={∭}_E\mathrm{ÒÏ}dxdydz$

$m={∭}_{E}k{z}^{2}dxdydz$

$m={∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}{∫}_{r=0}^{\cos \mathrm{θ}}{∫}_{z=r^2}^{\cos \mathrm{θ}}\left(k{z}^2\right)rdzdrd\mathrm{θ}$

$m={\left.{∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}{∫}_{r=0}^{\cos \mathrm{θ}}\left(kr\right)\left(\frac{z^3}{3}\right)\right|}_{z=r^2}^{z-r\cos \mathrm{θ}}drd\mathrm{θ}$

$m={∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}{∫}_{r=0}^{\cos \mathrm{θ}}\left(\frac{kr}{3}\right)\left(r^3{\cos }^3\mathrm{θ}-r^6\right)drd\mathrm{θ}$

Solving further yields

$m={\left.{∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\left(\frac{k}{3}\right)\left(\frac{r^5}{5}{\cos }^3\mathrm{θ}-\frac{r^8}{8}\right)\right|}_{r=0}^{\cos \mathrm{θ}}d\mathrm{θ}$

$m={\left(\frac{k}{3}\right)}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\left(\frac{{\cos }^8\mathrm{θ}}{5}-\frac{{\cos }^8\mathrm{θ}}{8}\right)d\mathrm{θ}$

$m=\left(\frac{2k}{3}\right)\left(\frac{3}{40}\right){∫}_{\mathrm{θ}=0}^{\mathrm{π}/2}{\cos }^8\mathrm{θ}d\mathrm{θ}$

Solve using integration

$m=\left(\frac{k}{20}\right){∫}_{\mathrm{θ}=0}^{\mathrm{π}/2}{\cos }^8\mathrm{θ}d\mathrm{θ}$

$m=\left(\frac{k}{20}\right)\left(\frac{7}{8}×\frac{5}{6}×\frac{3}{4}×\frac{1}{2}×\frac{\mathrm{π}}{2}\right)$

$m=\left(\frac{7k\mathrm{Ï€}}{1024}\right)$ (required mass)