The sum of the areas of rectangles formed over these subintervals. It is expressed as:
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0 ≤ sin x ≤ 1 and 1 ≤ 1+x² ≤ 1+(π/2)², but simpler: 0 ≤ f(x) ≤ 1 ⇒ 0 ≤ I ≤ π/2. Lower bound π/6 comes from sin x ≥ 2x/π? Accept as given.
\subsection*Solution 8 Rewrite: (\frac1n\sum_k=1^n \sin\left(\frack\pi2n\right) = \frac1\pi/2 \cdot \frac\pi2n\sum_k=1^n \sin\left(\frack\pi2n\right))? Actually: Let (\Delta x = \frac\pi/2n = \frac\pi2n), then the sum is (\frac1n\sum \sin(k\Delta x) = \frac2\pi\cdot \frac\pi2n\sum \sin(k\Delta x))? Wait: (\frac1n = \frac2\pi\cdot \frac\pi2n). So: [ \lim_n\to\infty \frac1n\sum_k=1^n \sin\left(\frack\pi2n\right) = \lim_n\to\infty \frac2\pi\sum_k=1^n \sin\left(\frack\pi2n\right)\cdot\frac\pi2n = \frac2\pi\int_0^\pi/2 \sin x,dx = \frac2\pi[-\cos x]_0^\pi/2 = \frac2\pi(0+1) = \frac2\pi. ]
The Riemann integral works by partitioning a closed interval
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