Area  Approximate the area under the graph of $f(x) =2x+1$ from $0$ to $4$ by finding $s_{n}$ and $S_{n}$ for $n = 4$ and $n = 8$.

Area  Approximate the area under the graph of $f(x) =x^{2}$ from 0 to 8 by finding $s_{n}$ and $S_{n}$ for $n = 4$ and $n = 8$ subintervals.

Area  Find the area $A$ under the graph of $y = f(x) = 9 - x^{2}$ from $0$ to $3$ by using lower sums $s_{n}$ (rectangles that lie below the graph of $f)$.

Area  Find the area $A$ under the graph of $y = f(x) = 8-2x$ from $0$ to $4$ using upper sums $S_{n}$ (rectangles that lie above the graph of $f).$

Riemann Sums

Units of an Integral  In the definite integral $\int_{a}^{b}a(t)\,dt,$ where $a$ represents acceleration measured in meters per second squared and $t$ is measured in seconds, what are the units of $\int_{a}^{b}a(t)\, dt?$

$\dfrac{d}{dx}\int_{0}^{x}t^{2/3}\sin t$ $dt$

$\dfrac{d}{dx}\int_{e}^{x}\ln t\, dt$

$\dfrac{d}{dx}\int_{x^{2}}^{1}\tan\, t\,dt$

$\dfrac{d}{dx}\int_{a}^{2\sqrt{x}}\dfrac{t}{t^{2}+1}\,dt$

$\int_{1}^{\sqrt{2}}x^{-2}\,dx$

$\int_{1}^{e^{2}}\dfrac{1}{x}\,dx$

$\int_{0}^{1}\dfrac{1}{1+x^{2}}\,dx$

$\int \dfrac{1}{x\sqrt{x^{2}-1}}\,dx$

$\int_{0}^{\ln 2}4e^{x}\,dx$

$\int_{0}^{2}( x^{2}-3x+2)~dx$

$\int_{1}^{4}2^{x}dx$

$\int_{0}^{\pi /4}\sec\, x\, \tan\, x\, dx$

$\int \left( \dfrac{1+2xe^{x}}{x}\right)\, dx$

$\int \dfrac{1}{2}\sin x~dx$

Interpreting an Integral  The function $v=v(t)$ is the speed $v,$ in kilometers per hour, of a train at a time $t$, in hours. Interpret the integral $\int_{0}^{16}v(t)\,dt=460.$

Interpreting an Integral  The function $V=f(t)$ is the volume $V$ of oil, in liters per hour, draining from a storage tank at time $t$ (in hours). Interpret the integral $\int_{0}^{2}f(t) dt=100.$

$\int_{-2}^{2}{f(x)\,dx},$ where ${f(x)=}{{\left\{ \begin{array}{c@{ }cc} 3x+2 & \hbox{if} & -2\leq x\lt0 \\ 2x^{2}+2 & \hbox{if} & 0\leq x\leq 2 \end{array} \right. }}$

$\int_{-1}^{4}\vert x\vert \,dx$

$\int_{-\pi /2}^{\pi /2}\sin x\,dx$

$\int_{-3}^{3}\dfrac{x^{2}}{x^{2}+9}\,dx$

$\int_{0}^{2}{(e^{x^2})\,dx }$

$\int_{0}^{1}\dfrac{1}{1+x^2}\,dx$

$\int_{0}^{\pi }{\sin x\,dx}$

$\int_{-3}^{3}\,(x^{3}+2x) dx$

$f(x) =\sin x$ over $\left[ -\dfrac{\pi }{2},\dfrac{\pi }{2}\right]$

$f(x) =x^{3}$ over $[1,4]$

$f(x) =e^{x}$ over $[-1,1]$

$f(x) =6x^{2/3}$ over $[ 0, 8 ]$

Find $\dfrac{d}{dx}\int \sqrt{\dfrac{1}{1+4x^{2}}}\,dx$

Find $\dfrac{d}{dx}\int\, \ln\, x\, dx$.

$\dfrac{dy}{dx}=3xy$;  $y=4$ when $x=0$

$\cos y\dfrac{dy}{dx}=\dfrac{\sin y}{x}$; $y=\dfrac{\pi }{3}$ when $x=-1$

$\int {{\dfrac{{y\,dy}}{{(y-2)^{3}}}}}$

$\int {{\dfrac{{x}}{{(2-3x)}^{3}}}\,dx}$

$\int {\sqrt{\dfrac{{1+x}}{{x^{5}}}}\,dx}$, $x>0$

$\int_{\pi ^{2}/4}^{4\pi ^{2}}\dfrac{1}{\sqrt{x}}\sin \sqrt{x}~dx$

$\int_{1}^{2}\dfrac{1}{t^{4}}\left( 1-\dfrac{1}{t^{3}}\right) ^{3}~dt$

$\int \dfrac{e^{x}+1}{e^{x}-1}dx$

$\int \dfrac{dx}{\sqrt{x}\,( 1-2\sqrt{x}) }$

$\int_{1/5}^{3}\dfrac{\ln (5x)}{x}~dx$

$\int_{-1}^{1}\dfrac{5^{-x}}{2^{x}}dx$

$\int e^{x+e^{x}}dx$

$\int_{0}^{1}\dfrac{x\,dx}{\sqrt{2-x^{4}}}$

$\int_{4}^{5}\dfrac{dx}{x\sqrt{x^{2}-9}}$

$\int {\sqrt[3]{{x^{3}+3\cos x}} (x^{2}-\sin x)\,dx}$

Find $f^{\prime \prime} (x)$ if $f(x)={\int_{0}^{x}\sqrt{1-t^{2}}}d{t}$.

Suppose that $F(x)={\int_{0}^{x}{\sqrt{t}\,dt}}$ and $G(x)={\int_{1}^{x}\sqrt{t}}\,dt$. Explain why $F(x)-G(x)$ is constant. Find the constant.

If $\int_{0}^{2}f(x + 2)\,dx\,=3$, find ${\int_{2}^{4}{f(x)\,dx.}}$

If $\int_{1}^{2}f(x - c)\,dx=5$, where $c$ is a constant, find $\int_{1-c}^{2-c}f(x)\,dx$.

Area  Find the area under the graph of $y=\cosh x$ from $x=0$ to $x=2$.

Water Supply  A sluice gate of a dam is opened and water is released from the reservoir at a rate of $r(t) =100+\sqrt{t}$ gallons per minute, where $t$ measures the time in minutes since the gate has been opened. If the gate is opened at 7 a.m. and is left open until 9:24 a.m., how much water is released?

Forensic Science  A body was found in a meat locker whose ambient temperature is $10{}^{\circ}{\rm C}.$ When the person was alive, his body temperature was $37{}^{\circ}{\rm C}$ and now it is $25{}^{\circ}{\rm C}.$ Suppose the rate of change of the temperature $u=u(t)$ of the body with respect time $t$ in hour (h) is proportional to $u(t)-T$, where $T$ is the ambient temperature and the constant of proportionality is $-0.294$.

Radioactive Decay  The amount $A$ of the radioactive element radium in a sample decays at a rate proportional to the amount of radium present. Given the half-life of radium is $1690$ years:

National Population Growth  Barring disasters (human-made or natural), the population $P$ of humans grows at a rate proportional to its current size. According to the U.N. World Population studies, from 2005 to 2010 the population of China grew at an annual rate of $0.510\%$ per year.