evaluate volume of paraboloid z = x2 + y2 between the planes z=0 and z=1
The Attempt at a Solution
i figured we would need to rearrange so that F(x,y,z) = x2 + y2 - z
then do a triple integral dxdydz of the function F. the limits for the first integral dz would be z=1 and z=0. and i dont know what the other limits would be (y1,y2 and x1, x2?)
but this first integral gave an answer of -1/2. this would mean that the volume would end up being zero which i dont think is right.
then i thought that maybe i should say the function is z(x,y) = x2 + y2 and integrating dxdy.
the dy limits would then be
[when z=1] y2 = 1 - x2
y = sqrt[1-x2]
[when z=0] y2 = -x2
y = sqrt[-x2] = xi i=complex number
the dx limits would then be
[when z=1] x2 = 1 - y2
x = sqrt[1-y2]
[when z=0] x2 = -y2
x = sqrt[-y2] = yi i=complex number
but this seems like a dead end
any suggestions would be helpful
ok just tried another thing: the lower limits for dx and dz being zero taking out the complex numbers.
this gives: after dy=> 2y dx = 2sqrt(1-x2)dx = x*sqrt(1-x2) + sin-1x
i then put in the limits 1 an 0 and got pi/2
still dont know if this is right though