Question

The base of a solid in the region bounded by the two parabolas y2 = 8x and x2 = 8y. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?

288 times pi over 35

576 times pi over 35

144 times pi over 35

8π

Answer 1

The two curves intersect at two points, (0, 0) and (8, 8):

`x^2=8y\implies y=\frac{x^2}8`

`y^2=(x^4)/(64)=8x\implies(x^4)/(64)-8x=0\implies(x(x-8)(x^2+8x+64))/(64)=0`

`\implies x=0,x=8\implies y=0,y=8`

The area of a semicircle with diameter
`d` is
`\frac{\pi d^2}8`. The diameter of each cross-section is determined by the vertical distance between the two curves for any given value of
`x` between 0 and 8. Over this interval,
`y^2=8x\implies y=√(8x)` and
`√(8x)>\frac{x^2}8`, so the volume of this solid is given by the integral

`\displaystyle\frac\pi8\int_0^8\left(√(8x)-\frac{x^2}8\right)^2\,\mathrm dx=(288\pi)/(35)`