I'm trying to evaluate the integral: $${\int_0}^1 {\int_0}^{\sqrt{1-x^2}} {\int_{\sqrt{x^2+y^2}}^{\sqrt{2-x^2-y^2}}} xy \space dxdydz$$
by converting to spherical coordinates.
I've got the equations of the surfaces:
$z^2 = x^2 + y^2$ $\space (1)$
$2=z^2 +x^2 +y^2$ $\space (2)$
$x=0$ $\space (3)$
$y=0$ $\space (4)$
and I want to convert these to spherical coordinates (in order for me to re-write the integral above as an integral with spherical coordinates).
I know that to convert to spherical coordinates:
$x=r \cos\rho \sin\theta$, $y=r \sin\rho \sin\theta$, $z=r \cos\theta$
So I substituted these values into my $4$ above equations but I was confused on how to simplify them.
I checked my answer sheet (which has no working out provided) and these are the answers they give:
$(1): r^2 \cos^2 \rho = r^2 \sin^2 \rho$
$(2): r=\sqrt 2$
$(3): \theta = \pi /2$
$(4): \theta = 0$
I have no idea how these were obtained with equations I have and the transformations I listed, maybe I did the simplification wrong but if anyone could show me how these spherical coordinates were obtained it would really help.
I think I subbed my $x,y,z$ values in and simplified wrong.
Thanks in advance
Intersection of $x^2+y^2+z^2=2$ and $z=\sqrt{x^{2}+y^{2}}$ on $OXY$ plane gives $x^2+y^2=1$, so we have circle in first quadrant.
Now let's take spherical coordinates: \begin{array}{} x = r \sin \phi \cos \theta; \\ y = r \sin \phi \sin \theta; \\ z = r \cos \phi \end{array}
From $z$ coordinate bounds $\sqrt{x^{2}+y^{2}} \leqslant z \leqslant \sqrt{2-x^{2}-y^{2}}$ we have $$r \sin \phi \leqslant r \cos \phi \leqslant \sqrt{2-r^2 \sin^2 \phi}$$
Left inequality gives $\sin \phi \leqslant \cos \phi$, from which we can obtain $\phi \leqslant \frac{\pi}{4}$. Right inequality gives $r \leqslant \sqrt{2}$. So for integral in spherical coordinates we have $$\int\limits_{0}^{\frac{\pi}{2}}\int\limits_{0}^{\frac{\pi}{4}}\int\limits_{0}^{\sqrt{2}} \underbrace{r^{2} \sin \theta}_{\text{Jacobian}} \cdot \underbrace{r \sin \phi \cos \theta}_{x} \cdot \underbrace{r \sin \phi \sin \theta}_{y} d r d \phi d \theta$$