Homework SourseEvaluate the volume enclosed by the paraboloid zx2y2 and the
Evaluate the volume enclosed by the paraboloid z=x^2+y^2 and the xy-plane over the cylinder x^2+y^2=2x.
Evaluate the volume enclosed by the paraboloid z=x^2+y^2 and the xy-plane over the cylinder x^2+y^2=2x.
Solution
The surface is the graph of the function f(x, y) = 2x 2 + 2y 2 over the region in the xy plane bounded by 2 = z = 2x 2 + 2y 2 and 8 = z = 2x 2 + 2y 2 . In polar coordinates, this is the region 1 r 2. The surface area is then Z 2 0 Z 2 1 q 1 + f 2 x + f 2 y r dr d = Z 2 0 Z 2 1 p 1 + (4x) 2 + (4y) 2 r dr d = Z 2 0 Z 2 1 p 1 + 16r 2 r dr d = Z 2 0 1 48 (1 + 16r 2 ) 3/2 r=2 r=1 d = Z 2 0 1 48 (653/2 173/2 ) d = 24 (653/2 173/2 ) .
