Calculate the integral of $f(x,y,w,z) = |y||z|^2$ over $S=\{(x,y,w,z)|x^2+y^2+z^2+w^2=1\}$, i.e. calculate: $\int_S|y||z|^2d\sigma_3$.
My attempt -
I'll define a parameterization as follows - $\phi(r,\alpha,\beta)=(rcos(\alpha),rsin(\alpha),rcos(\beta),rsin(\beta))$.
Using the following formula: $\int_Mf dS = \int\int\int f\circ \phi \sqrt{det(D_\phi*D_\phi^T)}drd\alpha d\beta$
Putting it all together I get that $\int_S|y||z|^2d\sigma_3 = \int_0^{2\pi}\int_0^{2\pi}\int_0^1|rsin(\alpha)|r^2sin(\beta)^2\sqrt{2}r^2drd\alpha d\beta = \frac{\sqrt{2}}{6}\pi\int_0^{2\pi}|sin(\alpha)|d\alpha = \frac{\sqrt{2}}{6}\pi(\int_0^\pi sin(\alpha) d\alpha - \int_\pi^{2\pi} sin(\alpha) d\alpha) = \frac{4\sqrt{2}}{6}\pi = \frac{2\sqrt{2}}{3}\pi$
Is my calculation correct?
And if not, what is the right way to calculate it?