Find the volume bounded by the paraboloid z 2x2 y2 and the

Solution

Find the volume bounded by the paraboloid z = 2x^2 + y^2 and the cylinder z = 4 - y^2. FIRST OCTANT. z=z z = 2x^2 + y^2 z = 4 - y^2. 2x^2 + y^2 = 4 - y^2. x^2+y^2 =2 circle V=\\int \\int( 4-y^2 -(2x^2 + y^2)) dA FIRST OCTANT. V/8 = \\int_{theta=0}^{pi/4} \\int_{r=0}^{sqrt(2)} ( 4- 2(r^2)) |J| dr d theta V/8 = \\int_{theta=0}^{pi/4} \\int_{r=0}^{sqrt(2)} ( 4- 2(r^2)) r dr d theta V/8 = pi/2