Question

Find the volume of the solid bounded by the graphs of the equations
z = x + y, x² + y² = 4 in the first octant.

Answer 1

   Visualize the solid.  It is like an inverted cone.  But the cone angle is not a constant.  The outer periphery of the solid (base of the cone like solid) is given by x² + y² = 2².  or,    r² = 2²

polar coordinates: 
    x= r cosФ   and  y = r sinФ 
    dA in a z plane :   (dr) (r dФ)
    The limits are    0<= r <= 2        and     0<= Ф <= π/2   for the first octant.
    dV = z dA = (x + y) dr  r dФ = r² (cosФ + sinФ) dr  dФ

Volume 
$= \int\limits^{r=2}_{r=0} {} \, {} \int\limits^{\pi/2}_{0} {r^2\ (sin\phi+cos\phi)} \, dr\ d\phi\\\\= \int\limits^{r=2}_{r=0} {r^2} \, {dr} \int\limits^{\pi/2}_{0} {(sin\phi+cos\phi)} \, d\phi\\\\=\frac{1}{3}[r^3]_0^2*[-cos\phi+sin\phi ]_0^{\pi/2}.\\\\=\frac{16}{3}.$

so the volume is 16/3