Question

The base of the solid S is the region bounded by the curves y = x 2 and y = 1, and cross-sections perpendicular to the x-axis are squares. Find the volume of S!

Answer 1

Each cross section has a side length equal to the distance (parallel to the
`y`-axis) between the curves
`y=1` and
`y=x^2`, which is
`1-x^2`, and hence area of
`(1-x^2)^2`. The two curves intersect at
`x=-1` and
`x=1`. Then the volume is equal to the integral

`\displaystyle\int_(-1)^1(1-x^2)^2\,\mathrm dx`

The integrand is even, since

`(1-(-x)^2)^2=(1-x^2)^2`

so we can make use of symmetry to simplify this to

`\displaystyle2\int_0^1(1-x^2)^2\,\mathrm dx`

Computing the integral is trivial:

`\displaystyle2\int_0^1(1-2x^2+x^4)\,\mathrm dx=2\left(x-\frac23x^3+\frac15x^5\right)\bigg|_0^1=\boxed{(16)/(15)}`