 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/6c/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Symmetry gives nodes

\pm c

with equal weights

b

. Exactness for

1

and

x^{2}

requires

2 b = \int_{- 1}^{1} x^{2} d x = \frac{2}{3}, 2 b c^{2} = \int_{- 1}^{1} x^{4} d x = \frac{2}{5} .

Therefore

b_{1} = b_{2} = \frac{1}{3}, c_{1} = - \frac{3}{5}, c_{2} = \frac{3}{5} .

Odd moments vanish automatically, so the rule is exact through degree three.

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