Note: (\sin(5\pi/8) = \sin(3\pi/8),\ \sin(7\pi/8) = \sin(\pi/8))
: (\int_0^2 x^2 dx = \fracx^33 \Big|_0^2 = \frac83 \approx 2.6667)
Exact: (\int_0^\pi \sin x , dx = 2). So (M_4 \approx 1.896) (error (\approx 0.104)). Express (\lim_n \to \infty \frac1n \sum_i=1^n \left(1 + \fracin\right)^3) as an integral.
: [ R_4 = 0.5 [f(0.5) + f(1) + f(1.5) + f(2)] = 0.5 [0.25 + 1 + 2.25 + 4] = 0.5 \times 7.5 = 3.75 ]
Sum: (\sum_i=0^n-1 4 = 4n,\ \sum_i=0^n-1 \frac6in = \frac6n \cdot \fracn(n-1)2 = 3(n-1))
[ M_4 \approx \frac\pi2 \times 1.306563 \approx 1.896 ]
Numerically: (\sin(22.5^\circ) \approx 0.382683,\ \sin(67.5^\circ) \approx 0.923880), sum (\approx 1.306563)
Exact: (\int_1^3 (3x+1)dx = \left[\frac3x^22 + x\right]_1^3 = \left(\frac272+3\right) - \left(\frac32+1\right) = (13.5+3)-(1.5+1)=16.5-2.5=14)
[ \int_a^b f(x) , dx = \lim_n \to \infty \sum_i=1^n f(x_i^*) \Delta x ]
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