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 \(\int_{a}^{b}a(t)\, dt?\)
In Problems 7–10, find each derivative using the Fundamental Theorem of Calculus.
\(\dfrac{d}{dx}\int_{0}^{x}t^{2/3}\sin t\) \(dt\)
\(\dfrac{d}{dx}\int_{e}^{x}\ln t\, dt\)
\(\dfrac{d}{dx}\int_{x^{2}}^{1}\tan\, t\,dt\)
\(\dfrac{d}{dx}\int_{a}^{2\sqrt{x}}\dfrac{t}{t^{2}+1}\,dt\)
In Problems 11–20, find each integral.
\(\int_{1}^{\sqrt{2}}x^{-2}\,dx\)
\(\int_{1}^{e^{2}}\dfrac{1}{x}\,dx\)
\(\int_{0}^{1}\dfrac{1}{1+x^{2}}\,dx\)
\(\int \dfrac{1}{x\sqrt{x^{2}-1}}\,dx \)
\(\int_{0}^{\ln 2}4e^{x}\,dx\)
\(\int_{0}^{2}( x^{2}-3x+2)~dx\)
\(\int_{1}^{4}2^{x}dx\)
\(\int_{0}^{\pi /4}\sec\, x\, \tan\, x\, dx \)
\(\int \left( \dfrac{1+2xe^{x}}{x}\right)\, dx\)
\(\int \dfrac{1}{2}\sin x~dx \)
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 \(\int_{0}^{2}f(t) dt=100.\)
In Problems 23–26, find each integral.
\(\int_{-2}^{2}{f(x)\,dx}, \) where \( {f(x)=}{{\left\{ \begin{array}{c@{ }cc} 3x+2 & \hbox{if} & -2\leq x\lt0 \\ 2x^{2}+2 & \hbox{if} & 0\leq x\leq 2 \end{array} \right. }}\)
\(\int_{-1}^{4}\vert x\vert \,dx \)
\(\int_{-\pi /2}^{\pi /2}\sin x\,dx\)
\(\int_{-3}^{3}\dfrac{x^{2}}{x^{2}+9}\,dx\)
Bounds on an Integral In Problems 27 and 28, find lower and upper bounds for each integral.
\(\int_{0}^{2}{(e^{x^2})\,dx } \)
\(\int_{0}^{1}\dfrac{1}{1+x^2}\,dx\)
In Problems 29 and 30, for each integral find the number(s) \(u\) guaranteed by the Mean Value Theorem for Integrals.
\(\int_{0}^{\pi }{\sin x\,dx}\)
\(\int_{-3}^{3}\,(x^{3}+2x) dx\)
In Problems 31–34, find the average value of each function over the given interval.
\(f(x) =\sin x\) 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 \(\dfrac{d}{dx}\int\, \ln\, x\, dx\).
In Problems 37 and 38, solve each differential equation using the given boundary condition.
\(\dfrac{dy}{dx}=3xy\); \(y=4\) when \(x=0\)
\(\cos y\dfrac{dy}{dx}=\dfrac{\sin y}{x}\); \(y=\dfrac{\pi }{3}\) when \(x=-1 \)
In Problems 39–51, find each integral.
\(\int {{\dfrac{{y\,dy}}{{(y-2)^{3}}}}}\)
\(\int {{\dfrac{{x}}{{(2-3x)}^{3}}}\,dx}\)
\(\int {\sqrt{\dfrac{{1+x}}{{x^{5}}}}\,dx} \), \(x>0\)
\(\int_{\pi ^{2}/4}^{4\pi ^{2}}\dfrac{1}{\sqrt{x}}\sin \sqrt{x}~dx\)
\(\int_{1}^{2}\dfrac{1}{t^{4}}\left( 1-\dfrac{1}{t^{3}}\right) ^{3}~dt\)
\(\int \dfrac{e^{x}+1}{e^{x}-1}dx\)
\(\int \dfrac{dx}{\sqrt{x}\,( 1-2\sqrt{x}) }\)
\(\int_{1/5}^{3}\dfrac{\ln (5x)}{x}~dx\)
\(\int_{-1}^{1}\dfrac{5^{-x}}{2^{x}}dx\)
\(\int e^{x+e^{x}}dx\)
\(\int_{0}^{1}\dfrac{x\,dx}{\sqrt{2-x^{4}}}\)
\(\int_{4}^{5}\dfrac{dx}{x\sqrt{x^{2}-9}}\)
\(\int {\sqrt[3]{{x^{3}+3\cos x}} (x^{2}-\sin x)\,dx}\)
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.
Source: U.N. World Population Prospects, 2010 update.