0-2-f-x-dx-f-a-f-2-a-For-what-values-of-a-is-the-following-formula-accurate-for-polynomials-of-degree-3-

Question Number 163300 by amin96 last updated on 05/Jan/22

\int_{0}^{2} f (x) d x ⋍ f (a) + f (2 - a)

For what values of a is the following formula accurate for polynomials of degree 3?

Answered by MJS_new last updated on 05/Jan/22

f (x) = c_{3} x^{3} + c_{2} x^{2} + c_{1} x + c_{0}

\underset{0}{\int^{2}} f (x) d x = f (a) + f (2 - a)

4 c_{3} + \frac{8}{3} c_{2} + 2 c_{1} + 2 c_{0} = 2 (3 c_{3} + c_{2}) a^{2} - 4 (3 c_{3} + c_{2}) a + 2 (4 c_{3} + 2 c_{2} + c_{1} + c_{0})

a^{2} - 2 a + \frac{2}{3} = 0

a = 1 \pm \frac{\sqrt{3}}{3}

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