Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$.

Question

Verify that the following formula is exact for polynomial of degree $≤ 4$: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$.

I'm not sure how to go about this. Any solutions/hints are greatly appreciated.

Answer 1

Hint. As integration and the right hand side are linear in $f$, the quadrature formula is exact for all polynomials of degree $\le 4$ iff it integrates the five functions $x \mapsto x^i$, $i = 0,\ldots, 4$ exactly. Just compute the right hand side values for those five $f$ and you are done.