91Ó°ÊÓ

The equations are$z=x$and$z=x^2+y^2$.

The relationship between rectangular and cylindrical coordinates are:

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

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

The rectangular coordinates are $z=x$and $z=x^2+y^2$

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

The cartesian limits are $x^2+y^2≤z≤x$

$x=r\cos \mathrm{θ}$

Hence, $z=x⇒z=r\cos \mathrm{θ}$

And $r^2=x^2+y^2$

Hence, $z=x^2+y^2⇒z=r^2$

It implies $r^2≤z≤r\cos \mathrm{θ}$

Simplifying

$x^2+y^2=x$

$⇒r^2=r\cos \mathrm{θ}$

Simplifying role="math" localid="1652376534758" $r=0,r=\cos \mathrm{θ}⇒0≤r≤\cos \mathrm{θ}$

The cylindrical limits are:

$r^2≤z≤r\cos \mathrm{θ},0≤r≤\cos \mathrm{θ}$&$-\mathrm{π}/2≤\mathrm{θ}≤\mathrm{π}/2$

Required volume is given by

$V={∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}{∫}_{r=0}^{\cos \mathrm{θ}}{∫}_{z=r^2}^{r\cos \mathrm{θ}}rdzdrd\mathrm{θ}$

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

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

$={\left.{∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\cos \mathrm{θ}\left(\frac{r^3}{3}\right)\right|}_{r=0}^{r=\cos \mathrm{θ}}d\mathrm{θ}-{\left.{∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\left(\frac{r^4}{4}\right)\right|}_{r=0}^{r=\cos \mathrm{θ}}d\mathrm{θ}$

$V={∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\frac{1}{3}\left({\cos }^4\mathrm{θ}\right)d\mathrm{θ}-{∫}_{\mathrm{θ}=-\mathrm{π}/2}^{\mathrm{π}/2}\frac{1}{4}\left({\cos }^4\mathrm{θ}\right)d\mathrm{θ}$

Further simplification gives

$V=\frac{2}{3}{∫}_{\mathrm{θ}=0}^{\mathrm{π}/2}{\cos }^4\mathrm{θ}d\mathrm{θ}-\frac{1}{4}·2{∫}_{\mathrm{θ}=0}^{\mathrm{π}/2}{\cos }^4\mathrm{θ}d\mathrm{θ}$

$V=\left(\frac{2}{3}\right)×\frac{3}{4}×\frac{1}{2}×\frac{\mathrm{π}}{2}-\left(\frac{1}{2}\right)×\frac{3}{4}×\frac{1}{2}×\frac{\mathrm{π}}{2}$

$V=\frac{4\mathrm{Ï€}}{32}-\frac{3\mathrm{Ï€}}{32}$

$V=\frac{\mathrm{Ï€}}{32}$units