Setting up the intergal but do not integrate

Question

I'm having a little trouble with this problem.

Let D be the solid bounded by y=x, z=1-y^2, x=0, and z=0

1) Sketch the region of integration using 2 and dimensional sketches to show the region clearly.

2). Setup 6 different integrals for calculating the volume of D, each with a different order of integration.

I can do number 1

but for 2 I am having trouble

I am thinking I need to setup intergals like this dydxdz dxdydz dzdydx dzdxdy dydzdx dxdzdy

If you could explain how to do go about doing this it will help a WHOLE bunch

Of course I don't want you guys to do my work for me but if you could do one order of integration and walk me through that as well it will help because my teacher isn't all that great.

Thank you much

Answer 1

Projecting $D$ on the plane $XY$: $$\text{The border of $D_z$ is given by}:\qquad x=y, 1-y^2=0, x=0.$$ The floor is $z=0$, the roof is $z=1-y^2$ and the triple integral can be written as $$\iint_{D_z}\int_0^{1-y^2}1\,dzdydx=\iint_{D_z}\int_0^{1-y^2}1\,dzdxdy.$$ Idem for the other projections on the coordinate planes.