Find the volume of the wedge in the first octant cut from the cylinder
$$ y^2+z^2=4 $$by the $yz$ plane and the plane $y=x$. Indicated in the figure the slice used to compute the volume.
I cant find the answer, please guide me on how to solve these type of questions.
The volume can be calculated using a triple integral:
$$\int\int_{D}\left(\int_0^{\sqrt{4-y^2}}dz\,\right)dxdy=\int\int_{D}\sqrt{4-y^2}\,dxdy$$
with the triangle $D=\{(x,y)\,|\,x<y\,,\,y<2\,,\,x>0\}$. Now
$$\int_0^2\int_0^y\sqrt{4-y^2}\,dxdy=\int_0^2y\sqrt{4-y^2}\,dy=\frac{8}{3}$$