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.
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.
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}$
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$