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

dwsmith

Well-known member
Feb 1, 2012
1,673
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

Administrator
Staff member
Feb 24, 2012
13,775
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
Feb 1, 2012
1,673
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

Administrator
Staff member
Feb 24, 2012
13,775
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:

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