Symmetric Matrices & Quadratic Forms · Superchallenge problem

Identify the polynomial $p(t) = a + bt + ct^2$ with the point $(a, b, c) \in \mathbb R^3$. Find the volume of the solid $\left\{(a, b, c) : \displaystyle\int_0^1 \big(a + bt + ct^2\big)^2\,dt \le 1\right\}$.