# [SOLVED]volume of a triangle type shape with a square bottom

#### dwsmith

##### Well-known member
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 c05.pdf

#### MarkFL

Staff member
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:

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

#### dwsmith

##### Well-known member
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

$$\displaystyle dV=\frac{1}{2}bh\,dx$$
where the base is a constant 1 and the height is $x$, hence:
$$\displaystyle dV=\frac{1}{2}x\,dx$$
$$\displaystyle V=1+\frac{1}{2}\int_0^1 x\,dx$$