To avoid stress concentrations, tunnels and support pillars
should be rounded. Find the **volume** of the square
cross-section pillar shown to the right.

We will use the geometry shown to the right. We will place the
parabola forming the left edge of the pillar with its axis of
symmetry on the *y* axis and its vertex at *y* = 3.
The first step is to find the equation of this parabola. The
parabola must have the form:

*y* = *a x*2 +
*b*,

where *b* is the *y* intercept (= 3) and *a* is a negative number.

*a* can be found by using the fact that the point (1,0) is
on the parabola:

*y* = *a x*2 + *b*,

*y* = *a x*2 + 3,

0 = *a* (1)2 +
3,

*a* = - 3,

Thus the equation of the parabola is:

*y* = - 3 *x*2 + 3.

The second step is to use this formula to find the dimensions of
the elemental rectangular slab shown in gray in the picture
above. The slab has its edge located at (*x*, *y*).
To find *x*, solve the above equation for
*x*:

Thus the volume of the rectangular slab element shown in gray is:

Integrating over all the elemental slabs from *y* = 0 to
*y* = 3 gives the entire volume of the pillar:

The volume of the pillar is 66 m3