Use polar coordinates to find the volume of the solid above the cone z = (x^2+y^2)^1/2 and below the hemisphere
z =( 8 − x^2 − y^2)^1/2.
You need to find
$\displaystyle V = \int \, \int_{R_{xy}} \sqrt{8 - x^2 - y^2} - \sqrt{x^2 + y^2} \, dx \, dy $
where $\displaystyle R_{xy}$ is the region inside the circle $\displaystyle x^2 + y^2 = 4$.
Note: $\displaystyle x^2 + y^2 = 8 - x^2 - y^2 \Rightarrow x^2 + y^2 = 4$.
Make the switch to polar coordinates:
$\displaystyle V = \int_{\theta = 0}^{\theta = 2 \pi} \int_{r = 0}^{r = 2} (\sqrt{8 - r^2} - r) \, r \, dr \, d\theta $
$\displaystyle = \int_{r = 0}^{r = 2} (\sqrt{8 - r^2} - r) \, r\, dr \cdot \int_{\theta = 0}^{\theta = 2 \pi} d \theta $.
Now do the simple integrations.