Question 5.114

According to Part 1 of the Fundamental Theorem of Calculus, ifa function \(f\) is continuous on a closed interval \([a,b] \), then \(\dfrac{d}{dx}\left[\int_{a}^{x}f(t)\, dt\right] =\) _________ for all numbers \(x\) in \((a,b)\).

Question 5.115

True or False  By Part 2 of the Fundamental Theorem of Calculus, \(\int_{a}^{b} x\, dx= b-a\).

Question 5.116

True or False  By Part 2 of the Fundamental Theorem of Calculus, \(\int_{a}^{b}f(x)\, dx=f(b)-f(a).\)

Question 5.117

True or False  \(\int_{a}^{b} F^\prime (x) dx\) can be interpreted as the rate of change in \(F\) from \(a\) to \(b.\)

Question 5.118

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

Question 5.119

\(\dfrac{d}{dx}\int_{3}^{x}\dfrac{t+1}{t}dt \)

Question 5.120

\(\dfrac{d}{dt}\left[\int_{0}^{t}{(3+x^{2})^{3/2}dx}\right] \)

Question 5.121

\(\dfrac{d}{dx}\left[\int_{-4}^{x}\left({t^{3}+8}\right)^{1/3}dt\right]\)

Question 5.122

\(\dfrac{d}{dx}\left[\int_{1}^{x}\ln u \,{du}\right]\)

Question 5.123

\(\dfrac{d}{dt}\left[\int_{4}^{t}e^{x}{dx}\right] \)

Question 5.124

\(\dfrac{d}{dx}\left[\int_{1}^{2x^{3}} \sqrt{t^{2}+1}\,dt \right] \)

Question 5.125

\(\dfrac{d}{dx}\left[\int_{1}^{\sqrt{x}}\sqrt{t^{4}+5} dt \right] \)

Question 5.126

\(\dfrac{d}{dx}\left[\int_{2}^{x^{5}}{\sec t} dt\right] \)

Question 5.127

\(\dfrac{d}{dx}\left[\int_{3}^{1/x}{\sin }^{5}t dt \right] \)

Question 5.128

\(\dfrac{d}{dx}\left[\int_{x}^{5}\sin ({t}^{2} ) \,dt \right] \)

Question 5.129

\(\dfrac{d}{dx}\left[\int_{x}^{3}{({t^{2}-5})^{10}\,dt} \right] \)

Question 5.130

\(\dfrac{d}{dx}\left[\int_{5x^{2}}^{5}(6t)^{2/3}\,dt \right] \)

Question 5.131

\(\dfrac{d}{dx}\left[\int_{x^{2}}^{0}e^{10t}\,dt \right]\)

Question 5.132

\(\int_{-2}^{3}{dx} \)

Question 5.133

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

Question 5.134

\(\int_{-1}^{2}{x^{3}dx} \)

Question 5.135

\(\int_{1}^{3}{\dfrac{1}{x^{3}}dx}\)

Question 5.136

\(\int_{0}^{1}\sqrt{u}\,du \)

Question 5.137

\(\int_{1}^{8}{\sqrt[3]{{y}}\,dy}\)

Question 5.138

\(\int_{\pi /6}^{\pi /2} \csc ^{2}{x\,dx} \)

Question 5.139

\(\int_{0}^{\pi /2}{\cos x\,dx}\)

Question 5.140

\(\int_{0}^{\pi /4}\sec x\tan x~dx\)

Question 5.141

\(\int_{\pi /6}^{\pi /2}\csc x\cot x\,dx\)

Question 5.142

\(\int_{-1}^{0}e^{x}dx \)

Question 5.143

\(\int_{-1}^{0}e^{-x}dx \)

Question 5.144

\(\int_{1}^{e}\dfrac{1}{x}dx\)

Question 5.145

\(\int_{e}^{1}\dfrac{1}{x}dx \)

Question 5.146

\(\int_{0}^{1}\dfrac{1}{1+x^{2}}dx \)

Question 5.147

\(\int_{0}^{\sqrt{2}/2}\dfrac{1}{\sqrt{1-x^{2}}}dx\)

Question 5.148

\(\int_{-1}^{8}x^{2/3}\,dx\)

Question 5.149

\(\int_{0}^{4}x^{3/2}\,dx\)

Question 5.150

Question 5.151

Question 5.152

Question 5.153

Question 5.154

Question 5.155

Question 5.156

Given that \(f(x)=(2x^{3}-3)^{2}\) and \(f^\prime (x)=12x^{2}(2x^{3}-3),\) find \(\int_{0}^{2}[12x^{2}(2x^{3}-3)]\, dx.\)

Question 5.157

Given that \(f(x)=(x^{2}+5)^{3}\) and \(f^\prime (x)=6x(x^{2}+5) ^{2},\) find \(\int_{-1}^{2}{6x(x^{2}+5)^{2}\, dx.}\)

Question 5.158

Area Find the area under the graph of \(f(x) =\dfrac{1}{\sqrt{1-x^{2}}}\) from \(0\) to \(\dfrac{1}{2}.\)

Question 5.159

Area Find the area under the graph of \(f(x) =\cosh x\) from \(-1\) to \(1\).

Question 5.160

Area Find the area under the graph of \(f(x)=\dfrac{1}{x^{2}+1}\) from \(0\) to \(\sqrt{3}.\)

Question 5.161

Area Find the area under the graph of \(f(x)=\dfrac{1}{1+x^{2}}\) from \(0\) to \(r\), where \(r>0\). What happens as \(r\rightarrow \infty \)?

Question 5.162

Area Find the area under the graph of \(y=\dfrac{1}{\sqrt{x}} \) from \(x=1\) to \(x=r\), where \(r>1\). Then examine the behavior of this area as \(r\rightarrow \infty \).

Question 5.163

Area Find the area under the graph of\(y=\dfrac{1}{x^{2}}\) from \(x=1\) to \(x=r\), where \(r>1\). Then examine the behavior of this area as \(r\rightarrow \infty \).

Question 5.164

Interpreting an Integral The function \(R=R(t) \) models the rate of sales of a corporation measured in millions of dollars per year as a function of the time \(t\) in years. Interpret the integral \(\int_{0}^{2}R(t)\,dt=23\).

Question 5.165

Interpreting an Integral The function \(v=v(t) \) models the speed \(v\) in meters per second of an object at atime \(t\) in seconds. Interpret the integral \(\int_{0}^{10}v(t)\,dt=4.8.\)

Question 5.166

Interpreting an Integral Helium is leaking from a large advertising balloon at a rate of \(H(t) \) cubic centimeters perminute, where \(t\) is measured in minutes.

1. Write an integral that models the change in helium in the balloonover the interval \(a\leq t\,\leq b.\)
2. What are the units of the integral from (a)?
3. Interpret \(\int_{0}^{300}H(t)\, dt=-100.\)

Question 5.167

Interpreting an Integral Water is being added to a reservoirat a rate of \(w(t) \) kiloliters per hour, where \(t\) is measured in hours.

1. Write an integral that models the change in amount of waterin the reservoir over the interval \(a\leq t\,\leq b.\)
2. What are the units of the integral from (a)?
3. Interpret \(\int_{0}^{36}w(t)\, dt=800.\)

Question 5.168

Free Fall  Thespeed \(v\) of an object dropped from rest is given by \(v(t)=9.8t\), where \(v\) is in meters per second and time \(t\) is inseconds.

1. Express the distance traveled in the first \(5.2\) s as an integral.
2. Find the distance traveled in \(5.2\) s.

Question 5.169

Area  Find \(h\) so that the area under the graph of \(y^{2}=x^{3}, 0\leq x\leq 4, y\geq 0\), is equal to the area of a rectangle of base \(4\) and height \(h\).

Question 5.170

Area  If \(P\) is apolynomial that is positive for \(x>0\), and for each \(k>0\) thearea under the graph of \(P\) from \(x=0\) to \(x=k\) is \(k^{3}+3k^{2}+6k\), find \(P\).

Question 5.171

Put It Together  If \(f(x)={\int_{0}^{x}}\dfrac{{1}}{\sqrt{t^{3}+2}}\,dt\), which of the following is false?

1. \(f\) is continuous at \(x\) for all \(x\geq 0\)
2. \(f(1)>0\)
3. \(f(0) =\dfrac{1}{\sqrt{2}}\)
4. \(f^\prime (1)=\dfrac{1}{\sqrt{3}}\)

Question 5.172

\(\int_{0}^{4}x^{2}dx\) and \(\int_{-4}^{4}x^{2}dx\)

Question 5.173

\(\int_{0}^{4}x^{3}dx\) and \(\int_{-4}^{4}x^{3}dx\)

Question 5.174

\(\int_{0}^{\pi /4}\sec ^{2}x\,dx\) and\(\int_{-\pi /4}^{\pi /4}\sec ^{2}x\,dx\)

Question 5.175

\(\int_{0}^{\pi /4}\sin x\,dx\) and \(\int_{-\pi/4}^{\pi /4}\sin x\,dx\)

Question 5.176

Area  Find \(c\),\(0\lt c\lt 1\), so that the area under the graph of \(y=x^2\) from 0 to \(c\) equals the area under the same graph from \(c\) to 1.

Question 5.177

Area Let \(A\) be the area under the graph of \(y=\dfrac{1}{x}\) from \(x=m\) to \(x=2m\), \(m>0\). Which of the following is true about the area \(A\)?

1. \(A\) is independent of \(m.\)
2. \(A\) increases as \(m\) increases.
3. \(A\) decreases as \(m\) increases.
4. \(A\) decreases as \(m\) increases when \(m\lt\dfrac{1}{2}\) and increases as \(m\) increases when \(m>\dfrac{1}{2}.\)
5. \(A\) increases as \(m\) increases when \(m\lt\dfrac{1}{2}\) anddecreases as \(m\) increases when \(m>\dfrac{1}{2}.\)

Question 5.178

Put It Together  If \(F\) is a function whose derivative is continuous for all real \(x\), find \[\lim\limits_{{_{h\rightarrow 0}}}{\dfrac{{1}}{{h}}}{\int_{c}^{c+h}{{F^\prime }(x)\,dx}}\]

Question 5.179

Suppose the closed interval \(\left[ 0,\dfrac{\pi }{2}\right] \)is partitioned into \(n\) subintervals, each of length \(\Delta x\), and \(u_{i}\)is an arbitrary number in the subinterval \([x_{i - 1},\,x_{i}],\) \(i\,=1,\,2,\,\ldots ,\,n\). Explain why \[\lim\limits_{{n\,\rightarrow \,\infty }}\,{\sum\limits_{i=1}^{n}}\left[ {{({\cos }\,u_{i})\,\Delta x}}\right] ={1}\]

Question 5.180

The interval \([0,4]\) is partitioned into \(n\) subintervals, eachof length \(\Delta x\), and a number \(u_{i}\) is chosen in thesubinterval \([x_{i-1},\,x_{i}],\) \(i=1,\,2,\,\ldots ,\,n\). Find \(\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}}( e^{u_1} \Delta x) .\)

Question 5.181

If \(u\) and \(\nu \) are differentiable functions and \(f\) is acontinuous function, find a formula for \[{\dfrac{{d}}{{dx}}}{\left[ {{\int_{u(x)}^{\nu (x)}{f(t)\,dt}}}\right] }\]

Question 5.182

Suppose that the graph of \(y=f(x)\) contains the points\((0,1)\) and \((2,5) \). Find \(\int_{0}^{2}f^\prime (x)\,dx\). (Assume that\(f^\prime \) is continuous.)

Question 5.183

If \(f^\prime \) is continuous on the interval \([a,b] \), show that\[\int_{a}^{b} f(x) f^\prime (x)\,dx=\dfrac{1}{2} \Big\{ [f(b)]^{2}-[f(a)]^{2} \Big\}.\]

[Hint: Look at the derivative of \(F(x) =\dfrac{\left[f(x) \right] ^{2}}{2}.]\)

Question 5.184

If \(f^{\prime \prime} \) is continuous on the interval \([a,b] \), show that\[\int_{a}^{b} xf^{\prime \prime} (x)\,dx=bf^\prime (b)-a f^\prime (a)-f(b)+f(a).\]

[Hint: Look at the derivative of \(F(x)=xf^\prime (x)-f(x).\)]

Question 5.185

What conditions on \(f\) and \(f^\prime\) guarantee that \(f(x)={\int_{0}^{x}{{f^\prime }(t)\,dt}}\)?

Question 5.186

Suppose that \(F\) is an antiderivative of \(f\) on the interval\([a,b] .\) Partition \([a,b] \) into \(n\)subintervals, each of length \(\Delta x_{i}=x_{i}-x_{i-1},\)\(i=1,2,\ldots,n\).

1. Apply the Mean Value Theorem for derivatives to \(F\) in eachsubinterval \([x_{i-1},x_{i}]\) to show that there is a point\(u_{i}\) in the subinterval for which \(F(x_{i})-F(x_{i-1})=f(u_{i})\Deltax_{i}\).
2. Show that\(\sum\limits_{i=1}^{n}[F(x_{i})-F(x_{i-1})]=F(b)-F(a).\)
3. Use parts (a) and (b) to explain why \[\int_{a}^{b}f(x)\,dx = F(b)-F(a).\](In this alternate proof of Part 2 of the FundamentalTheorem of Calculus, the continuity of \(f\) is not assumed.)

Question 5.187

Given \(y=\sqrt{x^{2} - 1} (4 - x),\) \(1\leq x\leq a\), forwhat number \(a\) will \(\int_{1}^{a}y\,dx\) have a maximum value?

Question 5.188

Find \(a>0\), so that the area under the graph of \(y=x+\dfrac{1}{x}\) from \(a\) to \((a+1) \) is minimum.

Question 5.189

If \(n\) is a known positive integer, for what number\(c\) is \[\int_{1}^{c}x^{n-1}\,dx=\dfrac{1}{n}\]

Question 5.190

Let \(f(x)={\int_{0}^{x}{{\dfrac{dt}{\sqrt{1-t^{2}}}}, 0 \lt x \lt1.}}\)

1. Find \(\dfrac{d}{dx} f(\sin x).\)
2. Is \(f\) one-to-one?
3. Does \(f\) have an inverse?