section_path: Area formulas/General formulas/Bounded area between two quadratic functions
section_name: Bounded area between two quadratic functions
paragraph_text: Bounded area between two quadratic functions
  
  To find the bounded area between two quadratic functions, we first subtract one from the other, writing the difference as f (x) − g (x) = a x 2 + b x + c = a (x − α) (x − β) {\displaystyle f(x)-g(x)=ax^{2}+bx+c=a(x-\alpha)(x-\beta)} where f (x) is the quadratic upper bound and g (x) is the quadratic lower bound. By the area integral formulas above and Vieta's formula, we can obtain that A = (b 2 − 4 a c) 3 / 2 6 a 2 = a 6 (β − α) 3, a ≠ 0. {\displaystyle A={\frac {(b^{2}-4ac)^{3/2}}{6a^{2}}}={\frac {a}{6}}(\beta -\alpha)^{3},\qquad a\neq 0.} The above remains valid if one of the bounding functions is linear instead of quadratic.