Question 1.

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 \((a,b)\).

Question 2.

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

Question 3.

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

Question 4.

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.

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

Question 6.

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

Question 7.

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

Question 8.

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

Question 9.

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

Question 10.

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

Question 11.

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

Question 12.

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

Question 13.

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

Question 14.

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

Question 15.

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

Question 16.

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

Question 17.

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

Question 18.

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

Question 19.

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

Question 20.

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

Question 21.

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

Question 22.

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

Question 23.

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

Question 24.

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

Question 25.

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

Question 26.

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

Question 27.

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

Question 28.

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

Question 29.

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

Question 30.

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

Question 31.

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

Question 32.

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

Question 33.

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

Question 34.

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

Question 35.

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

Question 36.

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

Question 37.

Question 38.

Question 39.

Question 40.

Question 41.

Question 42.

Question 43.

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 44.

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 45.

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

Question 46.

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

Question 47.

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

Question 48.

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 49.

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 50.

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 51.

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 52.

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 \(\int_{0}^{10}v(t)\,dt=4.8.\)

Question 53.

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 \(\int_{0}^{300}H(t)\, dt=-100.\)

Question 54.

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 \(\int_{0}^{36}w(t)\, dt=800.\)

Question 55.

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.

Question 56.

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 57.

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 \(P\).

Question 58.

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 59.

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

Question 60.

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

Question 61.

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

Question 62.

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

Question 63.

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 64.

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 \(m>\dfrac{1}{2}.\)

Question 65.

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 66.

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 67.

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 \(\lim\limits_{n\rightarrow \infty }{\sum\limits_{i=1}^{n}}( e^{u_1} \Delta x) .\)

Question 68.

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

Question 69.

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 70.

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 71.

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 72.

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

Question 73.

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.)

Question 74.

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?

Question 75.

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

Question 76.

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

Question 77.

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?