Setting triple integral boundaries

Question

I need to calculate the triple integral ∫∫∫y dxdydz over a region G which is given by {|x|<=z, 0<=z<=1, z<=y, x^2+y^2+z^2<=4}

I'm having trouble setting the boundaries for the integral

Would love an explanation, thanks in advance.

Answer 1

You have 5 surfaces that bound your region.

$x = z\\ x = -z\\ y= z\\ z = 0\\ z = 1$

and $x^2 + y^2 + z^2 = 4$

Here is a rough picture.

You need to integrate by y first or you else you will need to break this up.

$z \le y \le \sqrt {4-x^2 - z^2}$

And the rest is more direct.

$\int_0^1\int_{-z}^z\int_z^{\sqrt {4-x^2-z^2}} y \ dy \ dx\ dz$