[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 each of the following limits, find a function [math]f(x)[/math] such that the limit is equal to [math]\int_0^1 f(x) \; dx[/math]. Evaluate the limit.

• [math]\lim_{n\goesto\infty} \frac{1+2^2+3^2+\cdots+n^2}{n^3}[/math].
• [math]\lim_{n\goesto\infty} \frac{(1+n^2)+(2^2+n^2)+(3^2+n^2)+\cdots+(n^2+n^2)} {n^3}[/math].
• [math]\lim_{n\goesto\infty} \frac{\sqrt{1+n}+\sqrt{2+n}+\sqrt{3+n}+\cdots+\sqrt{n+n}} {n^{\frac32}}[/math].
• [math]\lim_{n\goesto\infty} \frac1{\sqrt n} \left(\frac1{\sqrt{1+n}}+\frac1{\sqrt{2+n}}+ \frac1{\sqrt{3+n}}+\cdots+\frac1{\sqrt{n+n}} \right)[/math].