Riemann Integral. Simpson's Rule. Approximation

Published 9/22/2025

\text{Evaluate:}\\ \text{(a) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sin\!\frac{k\pi}{n},\qquad \text{(b) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\tan\!\frac{k\pi}{3n}.\\[8pt] \textbf{Answer: } (a)\ \dfrac{2}{\pi};\quad (b)\ \dfrac{3}{\pi}\ln 2.\\[8pt] \textbf{Solution: }\\ \text{(a) Let }x_k=\tfrac{k\pi}{n},\ \Delta x=\tfrac{\pi}{n}.\ \frac{1}{n}\sum_{k=1}^{n}\sin\!\frac{k\pi}{n} =\frac{1}{\pi}\sum_{k=1}^{n}\sin(x_k)\,\Delta x\ \xrightarrow[n\to\infty]{}\ \frac{1}{\pi}\int_{0}^{\pi}\sin x\,dx=\frac{2}{\pi}.\\[6pt] \text{(b) Let }y_k=\tfrac{k\pi}{3n},\ \Delta y=\tfrac{\pi}{3n}.\ \frac{1}{n}\sum_{k=1}^{n}\tan\!\frac{k\pi}{3n} =\frac{3}{\pi}\sum_{k=1}^{n}\tan(y_k)\,\Delta y\ \xrightarrow[n\to\infty]{}\ \frac{3}{\pi}\int_{0}^{\pi/3}\tan x\,dx\\ =\frac{3}{\pi}\big[-\ln(\cos x)\big]_{0}^{\pi/3}=\frac{3}{\pi}\ln 2.

\text{Evaluate:}\\ \text{(a) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\cos^{2}\!\frac{k\pi}{n},\qquad \text{(b) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sec^{2}\!\frac{k\pi}{4n}.\\[6pt] \textbf{Answer: } (a)\ \dfrac{1}{2};\quad (b)\ \dfrac{4}{\pi}.

\text{Evaluate:}\\ \text{(a) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\!\left(\sin\!\frac{k\pi}{2n}\right)\!\left(\cos\!\frac{k\pi}{2n}\right),\quad \text{(b) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\sin^{2}\!\frac{k\pi}{2n}.\\[6pt] \textbf{Answer: } (a)\ \dfrac{1}{\pi};\quad (b)\ \dfrac{1}{2}.

\text{Evaluate:}\\ \text{(a) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\big|\sin\!\frac{2\pi k}{n}\big|,\qquad \text{(b) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\cos\!\frac{2\pi k}{n}.\\[6pt] \textbf{Answer: } (a)\ \dfrac{2}{\pi};\quad (b)\ 0.

\text{Evaluate:}\\ \text{(a) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\sin\!\frac{k\pi}{n}}{2+\cos\!\frac{k\pi}{n}},\qquad \text{(b) } \displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{1-\cos\!\frac{k\pi}{2n}}{1+\cos\!\frac{k\pi}{2n}}.\\[6pt] \textbf{Answer: } (a)\ \dfrac{1}{\pi}\ln 3;\quad (b)\ \dfrac{4}{\pi}-1.

\text{Composite trapezoidal rule on }[0,1]\text{ for }f(x)=x^2.\ \ h=\tfrac{1}{n}.\\[4pt] \text{(a) Write }T_n=h\Big[\tfrac{f(0)+f(1)}{2}+\sum_{k=1}^{n-1}f(kh)\Big].\\ \text{(b) Find a closed form for }T_n\text{ and the exact error }E_n=T_n-\int_{0}^{1}x^2\,dx.\\ \text{(c) Smallest }n\text{ with }|E_n|<10^{-3}.\\[8pt] \textbf{Answer: }T_n=\dfrac{2n^2+1}{6n^2},\quad \int_{0}^{1}x^2\,dx=\dfrac{1}{3},\quad E_n=\dfrac{1}{6n^2},\quad n_{\min}=13.\\[8pt] \textbf{Solution: }T_n=\tfrac{1}{n}\!\left[\tfrac{0+1}{2}+\sum_{k=1}^{n-1}\!\Big(\tfrac{k}{n}\Big)^2\right] =\tfrac{1}{n}\!\left[\tfrac{1}{2}+\tfrac{1}{n^2}\sum_{k=1}^{n-1}\!k^2\right].\\ \sum_{k=1}^{n-1}\!k^2=\tfrac{(n-1)n(2n-1)}{6}\ \Rightarrow\ T_n=\tfrac{1}{n}\!\left[\tfrac{1}{2}+\tfrac{(n-1)(2n-1)}{6n}\right]=\tfrac{2n^2+1}{6n^2}.\\ \int_{0}^{1}x^2dx=\tfrac{1}{3}\ \Rightarrow\ E_n=T_n-\tfrac{1}{3}=\tfrac{1}{6n^2}.\ |E_n|<10^{-3}\ \Rightarrow\ n^2>\tfrac{1}{6\cdot10^{-3}}=166.\overline{6}\ \Rightarrow\ n_{\min}=13.

\text{Composite trapezoidal rule on }[0,1]\text{ for }f(x)=x^3.\ \ h=\tfrac{1}{n}.\\[4pt] \text{(a) }T_n=h\Big[\tfrac{f(0)+f(1)}{2}+\sum_{k=1}^{n-1}f(kh)\Big].\quad \text{(b) }E_n=T_n-\int_{0}^{1}x^3dx.\quad \text{(c) Smallest }n\text{ with }|E_n|<10^{-3}.\\[8pt] \textbf{Answer: }T_n=\dfrac{n^2+1}{4n^2},\ \ \int_0^1 x^3dx=\dfrac{1}{4},\ \ E_n=\dfrac{1}{4n^2},\ \ n_{\min}=16.\\[8pt]

\text{Composite trapezoidal rule on }[0,2]\text{ for }f(x)=x^2.\ \ h=\tfrac{2}{n}.\\[4pt] \text{(a) Compute }T_n.\quad \text{(b) Exact }E_n=T_n-\int_0^2 x^2dx.\quad \text{(c) Smallest }n\text{ with }|E_n|<10^{-3}.\\[8pt] \textbf{Answer: }T_n=\dfrac{8}{3}+\dfrac{4}{3n^2},\ \ \int_0^2 x^2dx=\dfrac{8}{3},\ \ E_n=\dfrac{4}{3n^2},\ \ n_{\min}=37.\\[8pt]

\text{Composite trapezoidal rule on }[0,\pi]\text{ for }f(x)=\cos x.\ \ h=\tfrac{\pi}{n}.\\[4pt] \text{(a) Compute }T_n.\quad \text{(b) Exact }E_n=T_n-\int_{0}^{\pi}\cos x\,dx.\quad \text{(c) Smallest }n\text{ with }|E_n|<10^{-3}.\\[10pt] \textbf{Answer: }T_n=0,\ \ \int_{0}^{\pi}\cos x\,dx=0,\ \ E_n=0,\ \ n_{\min}=1.\\[10pt]

\text{Composite trapezoidal rule on }[0,1]\text{ for }f(x)=e^{x}.\ \ h=\tfrac{1}{n}.\\[4pt] \text{(a) Compute }T_n.\quad \text{(b) Exact }E_n=T_n-(e-1).\quad \text{(c) Smallest }n\text{ with }|E_n|<10^{-3}.\\[8pt] \textbf{Answer: }T_n=\dfrac{1}{n}\!\left[\dfrac{1+e}{2}+\dfrac{e-e^{1/n}}{e^{1/n}-1}\right],\ E_n=T_n-(e-1),\ n_{\min}=16.\\[8pt]