Using cylindrical polar coordinate

$x = r\cos θ, y = r\sin θ$

$dx\space dy\space dz = r\space dr\space dθ\space dz$

$z^2 = x^2 + y^2 = r^2\cos^2θ + r^2\sin^2θ$

$z^2 = x^2+y^2 = r^2 \to (1)$

Paraboloid $z = x^2 + y^2 = r^2 \to (2)$

Cone and parabola intersects in a circle

Substitute (2) in (1)

$(r^2)^2 = r^2$

$r^2 (r^2 - 1) = 0$

$r^2 = 0 \space (r^2-1) = 0$

limits of r are $r = 0$ to $r = 1$

or complete circle limits of θ are 0 to 2π .

Volume bounded by one and parabolaid

$Volume =\int\limits_{\theta=0}^{\theta=2\pi}\int\limits_{r=0}^{r=1}\int\limits_{z=r}^{z=r^2} r\space dr\space d\theta\space dz\\ =\int\limits_0^{2\pi}\int\limits_0^1r[r-r^2]dr\space d\theta\\ =\int\limits_0^{2\pi}\Bigg[\dfrac {r^3}4-\dfrac {r^4}4\Bigg]_0^1d\theta \\ =\int\limits_0^{2\pi}\Bigg(\dfrac 13-\dfrac 14\Bigg)d\theta \\ =\dfrac 1{12}[\theta]_0^{2\pi}\\ =\dfrac {2\pi}{12}=\dfrac \pi6$