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

A is correct

Explanation:

The base of a solid in the region bounded by the two parabolas
`y^2=8x` and
`x^2 = 8y`.

Two parabola equation:

`y_1=(x^2)/(8)`

`y_2=√(8x)`

Cross sections of the solid perpendicular to the x-axis are semicircles.

Diameter of semicircle
`=y_1-y_2=√(8x)-(x^2)/(8)`

Radius of semicircle (r)
`=√(2x)-(x^2)/(16)`

Thickness of solid
`=dx`

Range of
`x: 0\leq x \leq 8`

Volume of solid = Area of semicircle x Thickness

`V=\int_0^8(1)/(2)\pi\left ( √(2x)-(x^2)/(16) \right )^2dx\\\\V=\int_0^8(1)/(2)\pi\left ( 2x+(x^4)/(256)-(√(2)x^(3/2))/(8) \right )dx\\\\`

`V=(\pi)/(2)\left (x^2+(x^5)/(1280)-(2√(2)x^(5/2))/(40) \right )|_0^8`

`V=(\pi)/(2)((576)/(35)-0)`

`V=(288\pi)/(35)`

Hence, A is correct