Question 1.

If an interval \([a,b]\) is partitioned into \(n\) subintervals \([x_{0},\,x_{1}],\) \([x_{1},\,x_{2}],\) \([x_{2},\,x_{3}],\) \(\ldots, \) \([x_{n-1},\,x_{n}]\), where \(a=x_{0}\lt x_{1}\lt x_{2}\lt\cdots \lt x_{n-1}\lt x_{n}=b,\) then the set of subintervals of the interval \([a,b]\) is called a(n) ________ of \([a,b]\).

Question 2.

Multiple Choice  In a regular partition of \([0,40] \) into \(20\) subintervals, \( \Delta x= \) [(a)  20 (b)  40  (c)  2  (d)  4].

Question 3.

True or False  A function \(f\) defined on the closed interval \([a,b] \) has an infinite number of Riemann sums.

Question 4.

In the notation for a definite integral \(\int_{a}^{b}f(x)\, dx\), \(a\) is called the _______ _______; \(b\) is called the _______ _______; \(\int \) is called the _________ _________; and \(f(x) \) is called the_________.

Question 5.

If \(f(a) \) is defined, \(\int_{a}^{a}f(x)\, dx=\) _________.

Question 6.

True or False  If a function \(f\) is integrable over a closed interval \([a,b] \), then \(\int_{a}^{b}f(x) dx=\int_{b}^{a}f(x) dx\).

Question 7.

True or False  If a function \(f\) is continuous on a closed interval \([a,b] \), then the definite integral \(\int_{a}^{b}f(x)\, dx\) exists.

Question 8.

Multiple Choice  Since \(\int_{0}^{2}(3x-8) dx=-10\), then \(\int_{2}^{0}(3x-8)\, dx=\) [(a) \(-\)2  (b)  10  (c)  5  (d)  0].

Question 9.

\({f(x)=x}\), \(0\leq x \leq 2\). Partition the interval \([0,2]\) as follows: \[ \begin{eqnarray*} &&x_{0}=0, x_{1}=\dfrac{1}{4}, x_{2}=\dfrac{1}{2}, x_{3}=\dfrac{3}{4}, x_{4}=1, x_{5}=2;\\ &&\left[0,\dfrac{1}{4}\right], \left[\dfrac{1}{4},\dfrac{1}{2}\right], \left[\dfrac{1}{2},\dfrac{3}{4}\right], \left[\dfrac{3}{4},1\right], [1,2] \end{eqnarray*} \]

and choose \[ u_{1}=\dfrac{1}{8}, u_{2}=\dfrac{3}{8}, u_{3}=\dfrac{5}{8}, u_{4}=\dfrac{7}{8}, u_{5}=\dfrac{9}{8}. \]

Question 10.

\({f(x)=x}\), \(0\leq x \leq 2\). Partition the interval \([0,2]\) as follows: \(\left[0,\dfrac{1}{2}\right]\), \(\left[\dfrac{1}{2},1\right]\), \(\left[1,\dfrac{3}{2}\right]\), \(\left[\dfrac{3}{2},2\right]\), and choose \(u_{1}=\dfrac{1}{2}\), \(u_{2}=1\), \(u_{3}=\dfrac{3}{2}\), \(u_{4}=2\).

Question 11.

\(f(x)=x^{2}\), \(-2\leq x \leq 1\). Partition the interval \([-2,1]\) as follows: \([-2,-1]\), \([-1,0]\), \([0,1]\) and choose \(u_{1}={-}{\dfrac{3}{2}}\), \(u_{2}={-}{\dfrac{1}{2}}\), \(u_{3}={\dfrac{1}{2}}\).

Question 12.

\({f(x)=x^{2}}\), \(1\leq x\leq 2\). Partition the interval \([1,2]\) as follows: \(\left[1,\dfrac{5}{4}\right]\), \(\left[\dfrac{5}{4},\dfrac{3}{2}\right]\), \(\left[\dfrac{3}{2},\dfrac{7}{4}\right]\), \(\left[\dfrac{7}{4},2\right]\) and choose \(u_{1}=\dfrac{5}{4}\), \(u_{2}=\dfrac{3}{2}\), \(u_{3}=\dfrac{7}{4}\), \(u_{4}=2\).

Question 13.

Question 14.

Question 15.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}\left(e^{u_{i}}+2\right) \Delta x_{i}\) on \([0,2] \)

Question 16.

\(\lim\limits_{{\max \Delta x}_{i}\rightarrow 0}\sum\limits_{i=1}^{n}\ln u_{i}\Delta x_{i}\) on \([1,8]\)

Question 17.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}\cos u_{i}\Delta x_{i}\) on \([0,2\pi] \)

Question 18.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}\left( \cos u_{i}+\sin u_{i}\right) \Delta x_{i}\) on \([0,\pi] \)

Question 19.

\( \lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}{\dfrac{{2}}{{u^{2}_i}}}\Delta x_{i}\) on \([1,4] \)

Question 20.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}u^{1/3}_i\Delta x_{i}\) on \([0,8]\)

Question 21.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0}\sum\limits_{i=1}^{n}u_{i}\ln u_{i}\) \(\Delta x_{i}\) on \([1, e]\)

Question 22.

\(\lim\limits_{{\max \Delta x}_{i} \rightarrow 0} \sum\limits_{i=1}^{n}\ln (u_{i}+1)\Delta x_{i}\) on \([0, e] \)

Question 23.

\(\int_{-3}^{4}e\,dx \)

Question 24.

\(\int_{0}^{3}(-\pi) \,dx\)

Question 25.

\(\int_{3}^{0}(-\pi)\, dt\)

Question 26.

\(\int_{7}^{2}2\, ds\)

Question 27.

\(\int_{4}^{4}2\, \theta\ d\theta \)

Question 28.

\(\int_{-1}^{-1}8\, dr\)

Question 29.

Question 30.

Question 31.

Question 32.

Question 33.

\(\int_{0}^{\pi }\sin x\,dx\)

Question 34.

\(\int_{-\pi /4}^{\pi /4}\tan x\,dx\)

Question 35.

\(\int_{1}^{4}(x-2) ^{1/3}dx\)

Question 36.

\(\int_{1}^{4}(x+2) ^{1/3}dx\)

Question 37.

\(\int_{1}^{4}(\vert x \vert \,-2 ) \,dx\)

Question 38.

\(\int_{-2}^{4}\vert x \vert \,\,dx\)

Question 39.

\(f(x) =x^{2}-1\) on \([0,2]\)

Question 40.

\(f(x) =x^{3}-2\) on \([0,5]\)

Question 41.

\(f(x) =\sqrt{x+1}\) on \([0,3] \)

Question 42.

\(f(x) =\sin x\) on \([0, \pi] \)

Question 43.

\(f(x) =e^{x}\) on \([0, 2] \)

Question 44.

\(f(x) =e^{-x}\) on \([0,1] \)

Question 45.

\({\int_{0}^{1}(x - 4)dx} \)

Question 46.

\(\int_{0}^{3}{(3x - 1)dx}\)

Question 47.

\(f(x) =2+\sqrt{x}\) on \([1,5] \)

Question 48.

\(f(x) =e^{x}+e^{-x}\) on \([-1,3] \)

Question 49.

\(f(x) =\dfrac{3}{1+x^{2}}\) on \([-1,1] \)

Question 50.

\(f(x) = \dfrac{1}{\sqrt{x^2+4}} \) on \([0,2] \)

Question 51.

Find an approximate value of \({\int_{1}^{2}}\dfrac{1}{x}\,{dx}\) by finding Riemann sums corresponding to a partition of \([1,2]\) into four subintervals, each of the same length, and evaluating the integrand at the midpoint of each subinterval. Compare your answer with the true value, \(0.6931\ldots .\)

Question 52.

1. Find the approximate value of \({\int_{0}^{2} \sqrt{4 - x^{2}}}\,dx\) by finding Riemann sums corresponding to a partition of \([0, 2]\) into 16 subintervals, each of the same length, and evaluating the integrand at the left endpoint of each subinterval.
2. Can \(\int_{0}^{2}\sqrt{4-x^{2}}\, dx\) be interpreted as area? If it can, describe the area; if it cannot, explain why.
3. Find the actual value of \(\int_{0}^{2}\sqrt{4-x^{2}}dx\) by graphing \(y=\sqrt{4-x^{2}}\) and using a familiar formula from geometry.

Question 53.

Units of an Integral  In the definite integral \(\int_{0}^{5}F(x)\, dx\), \(F\) represents a force measured in newtons and \(x,\) \(0\leq x\leq 5,\) is measured in meters. What are the units of \(\int_{0}^{5}F(x)\, dx?\)

Question 54.

Units of an Integral  In the definite integral \(\int_{0}^{50}C(x)\, dx\), \(C\) represents the concentration of a drug in grams per liter and \(x,\) \(0\leq x\leq 50,\) is measured in liters of alcohol. What are the units of \(\int_{0}^{50}C(x)\, dx?\)

Question 55.

Units of an Integral  In the definite integral \(\int_{a}^{b}v(t)\, dt,\) \(v\) represents velocity measured in meters per second and time \(t\) is measured in seconds. What are the units of \(\int_{a}^{b}v(t)\, dt?\)

Question 56.

Units of an Integral  In the definite integral \(\int_{a}^{b}S(t)\, dt\), \(S\) represents the rate of sales of a corporation measured in millions of dollars per year and time \(t\) is measured in years. What are the units of \(\int_{a}^{b}S(t)\, dt?\)

Question 57.

Area

1. Graph the function \(f(x) =3-\sqrt{6x-x^{2}}.\)
2. Find the area under the graph of \(f\) from \(0\) to \(6.\)
3. Confirm the answer to (b) using geometry.

Question 58.

Area

1. Graph the function \(f(x) = \sqrt{4x-x^{2}}+2.\)
2. Find the area under the graph of \(f\) from \(0\) to \(4.\)
3. Confirm the answer to (b) using geometry.

Question 59.

The interval \([1,5]\) is partitioned into eight subintervals each of the same length.

1. What is the largest Riemann sum of \(f(x)=x^{2}\) that can be found using this partition?
2. What is the smallest Riemann sum?
3. Compute the average of these sums.
4. What integral has been approximated, and what is the integral's exact value?

Question 60.

The floor function \(f(x) = \) \(\lfloor x\rfloor \) is not continuous on \([0,4].\) Show that \({\int_{0}^{4}{f(x)\,dx}}\) exists.

Question 61.

Consider the Dirichlet function \(f,\) where \[ f(x)=\left\{ \begin{array}{@{}l@{ }l@{ }l} {1} & \hbox{if} & {x}~\hbox{is rational} \\ {0} & \hbox{if} & {x}~\hbox{is irrational} \end{array} \right. \] Show that \({\int_{0}^{1}{f(x)\,dx}}\) does not exist. (Hint: Evaluate the Riemann sums in two different ways: first by using rational numbers for \({ u}_{i}\) and then by using irrational number seak for \({ u}_{i}\).)

Question 62.

It can be shown (with a certain amount of work) that if \(f(x)\) is integrable on the interval \([a,b],\) then so is \(\vert f(x) \vert \). Is the converse true?

Question 63.

If only regular partitions are allowed, then we could not always partition an interval \([a,b]\) in a way that automatically partitions subintervals \([a,c] \) and \([c,b]\) for \(a\lt c\lt\,b\). Why not?

Question 64.

If \(f\) is a function that is continuous on a closed interval \([a,b] \), except at \(x_{1},\) \(x_{2},\) \( \ldots \), \(x_{n},\) \(n\geq 1\) an integer, where it has a jump discontinuity, show that \(f\) is integrable on \([a,b] .\)

Question 65.

If \(f\) is a function that is continuous on a closed interval \([a,b] \), except at \(x_{1},\) \(x_{2},\) \( \ldots \), \(x_{n},\) \(n\geq 1\) an integer, where it has a removable discontinuity, show that \(f\) is integrableon \([a,b] .\)