(a) show that the two integrals are equivalent and (b) calculate G2( f ). integral 0 to 2: sin(t) dt = integral -1 to 1:

[answerhappygod]

(a) show that the two integrals are equivalent and (b) calculate
G2( f ).
integral 0 to 2: sin(t) dt = integral -1 to 1: sin(x + 1) dx
Theorem 7.8: Let G2( f ) denote the two-point Gauss-Legendre
rule; If f is continuous on [−1, 1], then (13) 1 −1 f (x) dx ≈
G2( f ) = f −1 √3 + f 1 √3 . The Gauss-Legendre rule G2( f ) has
degree of precision n = 3. If f ∈ C4[−1, 1], then (14) 1 −1 f (x)
dx = f −1 √3 + f 1 √3 + E2( f ), where (15) E2( f ) = f (4) (c)
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Theorem 7.10 (Gauss-Legendre Translation). Suppose that the
abscissas{xN,k }N k=1 and weights {wN,k }N k=1 are given for the
N-point Gauss-Legendre rule over [−1, 1]. To apply the rule over
the interval [a, b], use the change of variable (20) t = a + b 2 +
b − a 2 x and dt = b − a 2 dx. Then the relationship (21) b a f (t)
dt = 1 −1 f a + b 2 + b − a 2 x b − a 2 dx is used to obtain the
quadrature formula (22) b a f (t) dt = b − a 2 N k=1 wN,k f a + b 2
+ b − a 2 xN,k