# Thread: volume of a triangle type shape with a square bottom

1. 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.

2. 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:

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

3. Originally Posted by 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:

$\displaystyle V=1+\frac{1}{2}\int_0^1 x\,dx=\frac{5}{4}$
How did you derive this formula? Is the 1 the volume of the cube or is that part of the triangular shape?

4. 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:

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

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