[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

(For those who have access to a high-speed digital computer and know how to use it.) Compute the Trapezoid approximation [math]T_n[/math] to each of the following integrals.

• [math]\int_0^1 \frac1{1+x^3} dx[/math], for [math]n=10[/math], [math]100[/math], and [math]1000[/math].
• [math]\int_0^1 \frac1{1+x^2} dx[/math], for successive values of [math]n = 10,100,1000,\ldots[/math], until the error is less that [math]10^{-6}[/math].
• [math]\int_0^1 \sqrt{1+x^3} \; dx[/math], for [math]n = 5[/math], [math]50[/math], and [math]500[/math].
• [math]\int_0^{\pi} \frac{\sin x}x dx[/math], for [math]n = 2[/math], [math]4[/math], [math]8[/math], [math]16[/math], and [math]100[/math].
• [math]\int_0^1 e^{-x^2}dx[/math], for [math]n=10[/math], [math]100[/math], and [math]1000[/math].