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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 \([a,b]\).

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

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

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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_________.

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

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

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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.

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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].

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\({f(x)=x}\), \(0\leq x \leq 2\). 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 \[ u_{1}=\dfrac{1}{8}, u_{2}=\dfrac{3}{8}, u_{3}=\dfrac{5}{8}, u_{4}=\dfrac{7}{8}, u_{5}=\dfrac{9}{8}. \]

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\({f(x)=x}\), \(0\leq x \leq 2\). 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 \(u_{1}=\dfrac{1}{2}\), \(u_{2}=1\), \(u_{3}=\dfrac{3}{2}\), \(u_{4}=2\).

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

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\({f(x)=x^{2}}\), \(1\leq x\leq 2\). 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 \(u_{1}=\dfrac{5}{4}\), \(u_{2}=\dfrac{3}{2}\), \(u_{3}=\dfrac{7}{4}\), \(u_{4}=2\).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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, \(0.6931\ldots .\)

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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.

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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 \(\int_{0}^{5}F(x)\, dx?\)

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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 \(\int_{0}^{50}C(x)\, dx?\)

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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 \(\int_{a}^{b}v(t)\, dt?\)

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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 \(\int_{a}^{b}S(t)\, dt?\)

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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.

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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.

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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?

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

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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 \({ u}_{i}\).)

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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?

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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?

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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 \([a,b] .\)

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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 \([a,b] .\)