Question

1 point) Find the limits of integration ly, uy, Ix, ux, lz, uz (some of which will involve variables z, y, z) so that dz dz dy represents the volume of the region in the first octant that is inside the paraboloid y z2 522 and between the planes y 1 and y -10. Ix uX uz ly

Answer 1

Inside the integral first dx is given so we have to find the limits of x

Given

$x=\pm \sqrt {y-5 z^{2}}$

Since the region is in the first octant x takes the limit $0 \leq x\leq \sqrt {y-5 z^{2}}$

Next to find limit of z :

Now,

$0\leq z \leq \sqrt{\frac{y}{5}}$

Any y takes the limit $1\leq y\leq 10$

Therefore,

$lx=0$

$ux=\sqrt{y-5 z^{2}}$

$lz=0$

1 Lz-

ly=1

.

uy = 10