Volume of a triangle type shape with a square bottom

[Dustinsfl]

How do I find the volume of this shape? The bottom is a square in the xy plane where \(0\leq x,y\leq 1\).

The object isn't a prism or pyramid so I am not sure what to do.

View attachment 1255

 

[MarkFL]

If I am interpreting this correctly, for $0\le z\le1$ you have a cube whose sieds are 1 unit in length, and for $1\le z\le2$ you have a solid whose cross-sections perpendicular to either the $x$ or $y$ axes are right triangles whose bases are 1 unit in length and altitudes vary linearly from 0 to 1, and so the volume by slicing is:

\(\displaystyle V=1+\frac{1}{2}\int_0^1 x\,dx=\frac{5}{4}\)

 

[Dustinsfl]

If I am interpreting this correctly, for $0\le z\le1$ you have a cube whose sieds are 1 unit in length, and for $1\le z\le2$ you have a solid whose cross-sections perpendicular to either the $x$ or $y$ axes are right triangles whose bases are 1 unit in length and altitudes vary linearly from 0 to 1, and so the volume by slicing is:

\(\displaystyle V=1+\frac{1}{2}\int_0^1 x\,dx=\frac{5}{4}\)

How did you derive this formula? Is the 1 the volume of the cube or is that part of the triangular shape?

 

[MarkFL]

Yes the 1 is the volume of the cubical portion of the solid, and for the upper part, the volume of a particular slice is:

\(\displaystyle dV=\frac{1}{2}bh\,dx\)

where the base is a constant 1 and the height is $x$, hence:

\(\displaystyle dV=\frac{1}{2}x\,dx\)

and so summing the slices (and adding in the cubical portion), we find:

\(\displaystyle V=1+\frac{1}{2}\int_0^1 x\,dx\)

 

[CellCoree]

To find the volume of this shape, you could use the formula for the volume of a pyramid, which is V = (1/3) * base area * height. In this case, the base area would be the area of the square, which is 1 * 1 = 1. The height of the shape can be determined by finding the distance between the base and the top of the triangle, which would be the length of the hypotenuse of the right triangle formed by the sides of the square and the height of the shape. Using the Pythagorean theorem, we can find the height to be √2. Plugging these values into the formula, we get V = (1/3) * 1 * √2 = √2/3, which is the volume of the triangle type shape with a square bottom.