Finite element methods involve integrals of functions that are not polynomials, and these integrals must be computed numerically.
For example, suppose that $f$ is the right-hand side of a Poisson problem and we use Lagrange elements of degree $k$. The right-hand side of the discrete linear system is computed from integrals
$\int_T f(x) p(x)$
where $p(x)$ is a polynomial of degree at most $k$ over the element $T$. What is the right way of computing that integral?
Use numerical quadrature for the integral. Considering the case $f = constant$ suggests that the quadrature formula should exactness at least $k$.
Replace the function $f(x)p(x)$ by a polynomial interpolant $\Pi[fp]$ and compute $\int_T \Pi[fp](x)$ exactly. Again, the interpolant should have at least degree $k$.
Replace the function $f(x)$ by a polynomial interpolant $\Pi[f](x)$ and compute $\int_T \Pi[f](x) p(x)$ exactly.
Both quadrature rules and interpolation require point evaluations of $f$. However, finite element textbooks always review quadrature rules. What are the considerations here?