How do you find the volume of this solid?

Volume = length * width * height

If z= 0 and z = x then x = 0, and if y = x, then y = 0 as well. I guess this problem is not correctly written.

I'll suppose that z = 0 is invisible, because if z = 0 then everything is 0 and there's no volume.

z = x , y = x , x+y = 3

y = x so x + x = 3

2x = 3

x = 3/2 = y = z

Then the volumen would be just:

V = 3/2 * 3/2 * 3/2

V = 3.375

As I see it, you actually have two solids that are stuck together. The first solid starts at (0,0,0) and expands as a vertical square along the x axis with the side of the square enlarging equal to x as x gets larger, ending at x=1.5. The second solid is a wedge or axe like shape that starts as a square with sides equal to 1.5 and tapering to a sharp vertical edge at x=3.

So the first solid can be calculated as the definite integral from 0 to 1.5 as ∫x²dx,

and the second solid is the definite integral from 1.5 to 3.0 of ∫x(3-x)dx.

I get 1.25 for the first volume and 8.25 for the second volume, so the total volume should be 9.5 cubic units. I hope that is correct.

Try to visualize the planes, solve for the intersection points.

Integrate as needed.