Find the exact volume of a solid whose base is bounded by the ellipse x^2+5y^2=1 and that has semicircular cross-sections taken perpendicular to the x-axis.

Answer 1

The elliptical base intersects the x-axis at x = 1 and x = -1.

The base is symmetric with respect to the y-axis. So, we can integrate from 0 to 1

and then multiply by 2.

The radius, y, of a typical semicircular cross section is y = √[(1 - x²) / 5]

Volume of typical cross section = (π/2)y²dx = (π/10)(1 - x²)dx

Volume of solid = 2∫_{(from 0 to 1)}[(π/10)(1 - x²)dx = (π/5)(x - (1/3)x³)_{(from 0 to 1)} = 2π/15