Functional integral equations

Let $f(x)$ be an integrable function such that:

$\int_{a}^{b} f(x) dx = F(x)|_{a}^{b} = F(b) - F(a) = 12.$

Turning to our double integral now yields:

$\int_{a}^{b} \int_{a}^{x} f(x)f(y) dy dx = \int_{a}^{b} f(x) [\int_{a}^{x} f(y) dy] dx = \int_{a}^{b} f(x)[F(x) - F(a)] dx;$

or $\frac{1}{2} \cdot F^{2}(x) - F(a)F(x)|_{a}^{b};$

or $\frac{1}{2}F^{2}(b) - \frac{1}{2}F^{2}(a) - F(a)F(b) + F^{2}(a);$

or $\frac{1}{2} \cdot (F(b) - F(a))^2 = \frac{12^2}{2} = \boxed{72}.$