Find the limits of integration ly, uy, lx, ux, lz, uz

Question

Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variable x,y,z) so that the triple integral from ly to uy, from lz to uz, from lx to ux, dx dz dy represents the volume of the region in the first octant that is inside the paraboloid y=x^2+4z^2 and between the planes y=5 and y=10. lx=___ ux=___ lz=___ uz=___ ly=___ uy=___

Answer 1

Since x^2 + 4z^2 <= y, lx = - sqrt(y - 4z^2), ux = sqrt(y - 4z^2), lz = -1/2 sqrt(y), uz = 1/2 sqrt(y), ly = 5, and uy = 10

If we made the transformation w = 2z, we could easily solve this problem using cylindrical coordinates.