Find The Volume

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?

Question options:

1)
2)
3)
4)	8π

Answer 1

Let find x-coordinate of intersection intersection points: y = x²/8 let put in second equation (x²/8)² = 8x,

x⁴ = 8³x, x = 0 and x = 8. Then diameter of section is equal (sqrt(8x) - x²/8). Then area of section is equal area of semicircle with this diameter: A(x) = π(sqrt(8x) - x²/8)²/4.

Then volume V = ∫₀⁸π(sqrt(8x) - x²/8)²/4dx = .π∫₀⁸(2x + x⁴/256 - 2^1/2 x^5/2/8⁾dx = π(x² + x⁵/1280 - 2^1/2x^7/2·2/(87))₀⁸= π(64 + 8⁵/1280 - 512/7) = 576π/35

Answer is 2) V = 576π/35