Finding Volume of Solid Bounded by Paraboloid and Planes

[Krayfish]

Homework Statement

Find the volume of the solid enclosed by the paraboloid z=x^2 + 3y^2 and the planes x=0, y=x, y=1, z=0

Homework Equations

I'm not really sure what's getting me about this, but I'm not really sure how to proceed after finding the x, y, and z intercepts...

The Attempt at a Solution

x intercept: 0
y intercept: 0
z intercept: 0

Would I just take bounds of y to be 1 and x and the bounds of x to be 0 and 1?
∫∫x^2 + 3y^2 dydx → The dy result would be yx^2 + y^3 → (x^3+x^3 -(x^2+1)) → 2X^3 - x^2 - 1 and then continue to integrate... ∫2x^3-x^2-1 dx → ((x^4)/2 -(x^2)/3 -x) → 1/2 - 1/3 - 1 = -5/6?

If that correct or am I missing something important?

 

[Mark44]

Homework Statement

Find the volume of the solid enclosed by the paraboloid z=x^2 + 3y^2 and the planes x=0, y=x, y=1, z=0

Homework Equations

I'm not really sure what's getting me about this, but I'm not really sure how to proceed after finding the x, y, and z intercepts...

The Attempt at a Solution

x intercept: 0
y intercept: 0
z intercept: 0

Would I just take bounds of y to be 1 and x and the bounds of x to be 0 and 1?
∫∫x^2 + 3y^2 dydx → The dy result would be yx^2 + y^3 → (x^3+x^3 -(x^2+1)) → 2X^3 - x^2 - 1 and then continue to integrate... ∫2x^3-x^2-1 dx → ((x^4)/2 -(x^2)/3 -x) → 1/2 - 1/3 - 1 = -5/6?

If that correct or am I missing something important?

Have you drawn a sketch of the solid whose volume you're supposed to find? Also, you should draw a sketch of the region in the x-y plane that forms the base of this solid. That would help you figure out what the integration limits need to be.

 

[Krayfish]

Have you drawn a sketch of the solid whose volume you're supposed to find? Also, you should draw a sketch of the region in the x-y plane that forms the base of this solid. That would help you figure out what the integration limits need to be.

I struggled to sketch it myself so I ended up using a software and plotted the parabaloid with the two y= planes. I'm not really sure where to go from there, all the examples we've done thus far have had at least 1 non zero intercept.