There is a theory of polynomials which enables
you to find the integral of a polynomial by
simply evaluating it at some special points and
adding certain multiples λ1,λ2,… of these values. For example,
for all polynomials q of degree less than six,
the special points are
−
√
3/5
, 0 and
+
√
3/5
, and
⌠ ⌡
1
−1
q(x)dx = λ1q(−
√
3/5
) + λ2q(0) +λ3q(+
√
3/5
).
(1)
Find the multiples λ1, λ2 and
λ3 by considering the three polynomials
q(x) = 1, q(x) = x and q(x) = x2.
With these values of λ1, λ2 and
λ3 show that the mechanical integration
given by equation (1), which uses the values of
the polynomial at the three special points, gives
the value of the integral of ALL quadratic
polynomials.
Now go on to show that the same formula gives the
integral of ALL cubic, quartic and quintic
polynomials.
Does the formula (1) hold for q(x)=x6?