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According to Part 1 of the Fundamental Theorem of Calculus, if a 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 \((a,b)\).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 \(r\rightarrow \infty \)?

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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 \(r\rightarrow \infty \).

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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 \(r\rightarrow \infty \).

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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 \(\int_{0}^{2}R(t)\,dt=23\).

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

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

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

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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 water in the reservoir over the interval \(a\leq t\,\leq b.\)
2. What are the units of the integral from (a)?
3. Interpret \(\int_{0}^{36}w(t)\, dt=800.\)

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

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

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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 \(h\).

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

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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. \(f^\prime (1)=\dfrac{1}{\sqrt{3}}\)

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

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

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

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

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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.

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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 \(m\lt\dfrac{1}{2}\) and decreases as \(m\) increases when \(m>\dfrac{1}{2}.\)

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

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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 \[ \lim\limits_{{n\,\rightarrow \,\infty }}\,{\sum\limits_{i=1}^{n}}\left[ {{({ \cos }\,u_{i})\,\Delta x}}\right] ={1} \]

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

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

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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.)

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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 \(F(x) =\dfrac{\left[f(x) \right] ^{2}}{2}.]\)

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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 \(F(x)=xf^\prime (x)-f(x).\)]

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

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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 each subinterval \([x_{i-1},x_{i}]\) to show that there is a point \(u_{i}\) in the subinterval for which \(F(x_{i})-F(x_{i-1})=f(u_{i})\Delta x_{i}\).
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 Fundamental Theorem of Calculus, the continuity of \(f\) is not assumed.)

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

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

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

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

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