Finding Limits for Triple Integrals: How to Solve for the Intersection of Planes

[Timebomb3750]

Homework Statement

Use a triple integral to find the volume of the region. Below x+2y+2z=4, above z=2x, in the first octant.

Homework Equations

V=∫∫∫dV=∫∫∫dxdydz

The Attempt at a Solution

I have no clue where to begin as to finding those darn limits to integrate with. I'm sure I can evaluate the integral just fine, but I need help finding limits.

 

[Timebomb3750]

Well, after plotting those two equations into my mac grapher app, it seems my y-limits could be from 0 to 2. But I'm unsure as to finding my x and z limits.

 

[Timebomb3750]

Any assistance would be greatly appreciated. Thanks.

 

[SammyS]

Any assistance would be greatly appreciated. Thanks.

Patience, please.

(Look at the rules for posting on this Forum, especially as regards "bumping" your thread.)

 

[SammyS]

Homework Statement

Use a triple integral to find the volume of the region. Below x+2y+2z=4, above z=2x, in the first octant.

Homework Equations

V=∫∫∫dV=∫∫∫dxdydz

The Attempt at a Solution

I have no clue where to begin as to finding those darn limits to integrate with. I'm sure I can evaluate the integral just fine, but I need help finding limits.

Well, after plotting those two equations into my mac grapher app, it seems my y-limits could be from 0 to 2. But I'm unsure as to finding my x and z limits.

Where do the planes, x+2y+2z=4, and, z=2x, intersect?

Where does each of those planes intersect the coordinate axes?

 

[Timebomb3750]

Where do the planes, x+2y+2z=4, and, z=2x, intersect?

Where does each of those planes intersect the coordinate axes?

Well, z=2x goes through the entire the y-axis, but doesn't intersect any other axes. x+2y+2z=4 intersects axes at x=4, y=2, and z=2.

The two planes intersect at 2y+5x=4.

But what's your point? What do I get out of this?

 

[SammyS]

Well, z=2x goes through the entire the y-axis, but doesn't intersect any other axes. x+2y+2z=4 intersects axes at x=4, y=2, and z=2.

The two planes intersect at 2y+5x=4.

But what's your point? What do I get out of this?

I should have asked, "Where does each of those planes intersect the coordinate planes?"

The intersection of the two planes is a line. The equation, 2y+5x=4, specifies a plane !