Prove that `pi/6

Answer

NA

Explanation

Since `0ltx^(3)ltx^(2)`, we have (for `0ltxlt1`)
`x^(2)lt x^(2)+x^(3)lt2x^(2)`
or `-2x^(2)lt-x^(2)-x^(3)lt-x^(2)`
or `4-2x^(2)lt4-x^(2)-x^(3)lt4-x^(2)`
or `sqrt(4-2x^(2))ltsqrt(4-x^(2)-x^(3))ltsqrt(4-x^(2))`
or `1/(sqrt(4-x^(2)))lt 1/(sqrt(4-x^(2)-x^(3)))lt 1/(sqrt(4-2x^(2)))`
or `int_(0)^(1)1/(sqrt(4-x^(2)))dx lt int_(0)^(1)1/(sqrt(4-x^(2)-x^(3)))dx lt int_(0)^(1)1/(sqrt(4-2x^(2)))dx`
or `sin^(-1)(x/2)]_(0)^(1)lt int_(0)^(1) (dx)/(sqrt(4-x^(2)-x^(3))) lt 1/(sqrt(2))"sin"^(-1)x/(sqrt(2))]_(0)^(1)`
or `(pi)/6 lt int_(0)^(1)(dx)/(sqrt(4-x^(2)-x^(3)))lt (pi)/(4sqrt(2))`