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Mathematics LibreTexts

5.1 Approximating Area (Riemann Sum) Exercises

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    13754
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    These are homework exercises to accompany OpenStax's "Calculus" Textmap.

    5.1: Approximating Areas

    1.) State whether the given sums are equal or unequal.

    1. \(\displaystyle \sum_{i=1}^{10}i\) and \(\displaystyle \sum_{k=1}^{10}k\)
    2. \(\displaystyle \sum_{i=1}^{10}i\) and \(\displaystyle \sum_{i=6}^{15}(i−5)\)
    3. \(\displaystyle \sum_{i=1}^{10}i(i−1)\) and \(\displaystyle \sum_{j=0}^9(j+1)j\)
    4. \(\displaystyle \sum_{i=1}^{10}i(i−1)\) and \(\displaystyle \sum_{k=1}^{10}(k^2−k)\)
    Answer:
    a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting \(\displaystyle j=i−1.\) d. They are equal; the first sum factors the terms of the second.

    In the following exercises, compute the sums. (You can use the rules for sums of powers of integers.)

    2) \(\displaystyle \sum_{i=5}^{10}i\)

    3) \(\displaystyle \sum_{i=5}^{10}i^2\)

    Answer:
    \(\displaystyle 355\)

    Suppose that \(\displaystyle \sum_{i=1}^{100}a_i=15\) and \(\displaystyle \sum_{i=1}^{100}b_i=−12.\) In the following exercises, compute the sums.

    4) \(\displaystyle \sum_{i=1}^{100}(a_i+b_i)\)

    5) \(\displaystyle \sum_{i=1}^{100}(a_i−b_i)\)

    Answer:
    \(\displaystyle 15−(−12)=27\)

    6) \(\displaystyle \sum_{i=1}^{100}(3a_i−4b_i)\)

    7) \(\displaystyle \sum_{i=1}^{100}(5a_i+4b_i)\)

    Answer:
    \(\displaystyle 5(15)+4(−12)=27\)
     

    In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

    8) \(\displaystyle \sum_{k=1}^{20}100(k^2−5k+1)\)

    9) \(\displaystyle \sum_{j=1}^{50}(j^2−2j)\)

    Answer:
    \(\displaystyle \sum_{j=1}^{50}j%2−2\sum_{j=1}^{50}j=\frac{(50)(51)(101)}{6}−\frac{2(50)(51)}{2}=40, 375\)

    10) \(\displaystyle \sum_{j=11}^{20}(j^2−10j)\)

    11) \(\displaystyle \sum_{k=1}^{25}[(2k)^2−100k]\)

    Answer:
    \(\displaystyle 4\sum_{k=1}^{25}k^2−100\sum_{k=1}^{25}k=\frac{4(25)(26)(51)}{9}−50(25)(26)=−10, 400\)

     

    Let \(\displaystyle L_n\) denote the left-endpoint sum using n subintervals and let \(\displaystyle R_n\) denote the corresponding right-endpoint sum. In the following exercises, compute the indicated left and right sums for the given functions on the indicated interval.

    12) \(\displaystyle L_4\) for \(\displaystyle f(x)=\frac{1}{x−1}\) on \(\displaystyle [2,3]\)

    13) \(\displaystyle R_4\) for \(\displaystyle g(x)=cos(πx)\) on \(\displaystyle [0,1]\)

    Answer:
    \(\displaystyle R_4= - 0.25\)

    14) \(\displaystyle L_6\) for \(\displaystyle f(x)=\frac{1}{x(x−1)}\) on \(\displaystyle [2,5]\)

    15) \(\displaystyle R_6\) for \(\displaystyle f(x)=\frac{1}{x(x−1)}\) on \(\displaystyle [2,5]\)

    Answer:
    \(\displaystyle R_6=0.372\)

    16) \(\displaystyle R_4\) for \(\displaystyle \frac{1}{x^2+1}\) on \(\displaystyle [−2,2]\)

    17) \(\displaystyle L_4\) for \(\displaystyle \frac{1}{x^2+1}\) on \(\displaystyle [−2,2]\)

    Answer:
    \(\displaystyle L_4=2.20\)

    18) \(\displaystyle R_4\) for \(\displaystyle x^2−2x+1\) on \(\displaystyle [0,2]\)

    19) \(\displaystyle L_8\) for \(\displaystyle x^2−2x+1\) on \(\displaystyle [0,2]\)

    Answer:
    \(\displaystyle L_8=0.6875\)

    20) Compute the left and right Riemann sums— \(\displaystyle L_4\) and \(\displaystyle R_4\), respectively—for \(\displaystyle f(x)=(2−|x|)\) on \(\displaystyle [−2,2].\) Compute their average value and compare it with the area under the graph of f.

    21) Compute the left and right Riemann sums— \(\displaystyle L_6\) and \(\displaystyle R_6\), respectively—for \(\displaystyle f(x)=(3−|3−x|)\) on \(\displaystyle [0,6].\) Compute their average value and compare it with the area under the graph of f.

    Answer:
    \(\displaystyle L_6=9.000=R_6\). The graph of f is a triangle with area 9.

    22) Compute the left and right Riemann sums— \(\displaystyle L_4\) and \(\displaystyle R_4\), respectively—for \(\displaystyle f(x)=\sqrt{4−x^2}\) on \(\displaystyle [−2,2]\) and compare their values.

    23) Compute the left and right Riemann sums— \(\displaystyle L_6\) and \(\displaystyle R_6\), respectively—for \(\displaystyle f(x)=\sqrt{9−(x−3)^2}\) on \(\displaystyle [0,6]\) and compare their values.

    Answer:
    \(\displaystyle L_6=13.12899=R_6\). They are equal.

     

    Express the following endpoint sums in sigma notation but do not evaluate them.

    24) \(\displaystyle L_{30}\) for \(\displaystyle f(x)=x^2\) on \(\displaystyle [1,2]\)

    25) \(\displaystyle L_{10}\) for \(\displaystyle f(x)=\sqrt{4−x^2}\) on \(\displaystyle [−2,2]\)

    Answer:
    \(\displaystyle L_{10}=\frac{4}{10}\sum_{i=1}^{10}\sqrt{4−(−2+4\frac{(i−1)}{10})}\)

    26) \(\displaystyle R_{20}\) for \(\displaystyle f(x)=sinx\) on \(\displaystyle [0,π]\)

    27) \(\displaystyle R_{100}\) for \(\displaystyle lnx\) on \(\displaystyle [1,e]\)

    Answer:
    \(\displaystyle R_{100}=\frac{e−1}{100}\sum_{i=1}^{100}ln(1+(e−1)\frac{i}{100})\)

     

    In the following exercises, graph the function then use a calculator or a computer program to evaluate the following left and right endpoint sums. Is the area under the curve between the left and right endpoint sums?

    28) [T] \(\displaystyle L_{100}\) and \(\displaystyle R_{100}\) for \(\displaystyle y=x^2−3x+1\) on the interval \(\displaystyle [−1,1]\)

    29) [T] \(\displaystyle L_{100}\) and \(\displaystyle R_{100}\) for \(\displaystyle y=x^2\) on the interval \(\displaystyle [0,1]\)

     

    Answer:
    \(\displaystyle R_{100}=0.33835,L_{100}=0.32835.\) The plot shows that the left Riemann sum is an underestimate because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.

    30) [T] \(\displaystyle L_{50}\) and \(\displaystyle R_{50}\) for \(\displaystyle y=\frac{x+1}{x^2−1}\) on the interval \(\displaystyle [2,4]\)

    31) [T] \(\displaystyle L_{100}\) and \(\displaystyle R_{100}\) for \(\displaystyle y=x^3\) on the interval \(\displaystyle [−1,1]\)

    Answer:
    \(\displaystyle L_{100}=−0.02,R_{100}=0.02\). The left endpoint sum is an underestimate because the function is increasing. Similarly, a right endpoint approximation is an overestimate. The area lies between the left and right endpoint estimates.

    32) [T] \(\displaystyle L_{50}\) and \(\displaystyle R_{50}\) for \(\displaystyle y=tan(x)\) on the interval \(\displaystyle [0,\frac{π}{4}]\)

    33) [T] \(\displaystyle L_{100}\) and \(\displaystyle R_{100}\) for \(\displaystyle y=e^{2x}\) on the interval \(\displaystyle [−1,1]\)

    Answer:
    \(\displaystyle L_{100}=3.555,R_{100}=3.670\). The plot shows that the left Riemann sum is an underestimate because the function is increasing. Ten rectangles are shown for visual clarity. This behavior persists for more rectangles.

    34) Let \(\displaystyle t_j\) denote the time that it took Tejay van Garteren to ride the jth stage of the Tour de France in 2014. If there were a total of 21 stages, interpret \(\displaystyle \sum_{j=1}^{21}t_j\).

    35) Let \(\displaystyle r_j\) denote the total rainfall in Portland on the jth day of the year in 2009. Interpret \(\displaystyle \sum_{j=1}^{31}r_j\).

    Answer:
    The sum represents the cumulative rainfall in January 2009.

    36) Let \(\displaystyle d_j\) denote the hours of daylight and \(\displaystyle δ_j\) denote the increase in the hours of daylight from day \(\displaystyle j−1\) to day j in Fargo, North Dakota, on thejth day of the year. Interpret \(\displaystyle d1+\sum_{j=2}^{365}δ_j\).

    37) To help get in shape, Joe gets a new pair of running shoes. If Joe runs 1 mi each day in week 1 and adds \(\displaystyle \frac{1}{10}\) mi to his daily routine each week, what is the total mileage on Joe’s shoes after 25 weeks?

    Answer:
    The total mileage is \(\displaystyle 7×\sum_{i=1}^{25}(1+\frac{(i−1)}{10})=7×25+\frac{7}{10}×12×25=385mi\).

    38) The following table gives approximate values of the average annual atmospheric rate of increase in carbon dioxide (CO2) each decade since 1960, in parts per million (ppm). Estimate the total increase in atmospheric CO2 between 1964 and 2013.

    Decade Ppm/y
    1964-1973 1.07
    1976-1983 1.34
    1984-1993 1.40
    1994-2003 1.87
    2004-2013 2.07

    Average Annual Atmospheric CO2 Increase, 1964–2013  Source: http://www.esrl.noaa.gov/gmd/ccgg/trends/.

     

    39) The following table gives the approximate increase in sea level in inches over 20 years starting in the given year. Estimate the net change in mean sea level from 1870 to 2010.

    Starting Year 20- Year Change
    1870 0.3
    1890 1.5
    1910 0.2
    1930 2.8
    1950 0.7
    1970 1.1
    1990 1.5

    Approximate 20-Year Sea Level Increases, 1870–1990

    Source: http://link.springer.com/article/10....712-011-9119-1

    Answer:
    Add the numbers to get 8.1-in. net increase.
     

    40) The following table gives the approximate increase in dollars in the average price of a gallon of gas per decade since 1950. If the average price of a gallon of gas in 2010 was $2.60, what was the average price of a gallon of gas in 1950?

    Starting Year 10- Year Change
    1950 0.03
    1960 0.05
    1970 0.86
    1980 −0.03
    1990 0.29
    2000 1.12

    Approximate 10-Year Gas Price Increases, 1950–2000

    Source: http://epb.lbl.gov/homepages/Rick_Di...011-trends.pdf.

    41) The following table gives the percent growth of the U.S. population beginning in July of the year indicated. If the U.S. population was 281,421,906 in July 2000, estimate the U.S. population in July 2010.

    Year % Change/Year
    2000 1.12
    2001 0.99
    2002 0.93
    2003 0.86
    2004 0.93
    2005 0.93
    2006 0.97
    2007 0.96
    2008 0.95
    2009 0.88

    Annual Percentage Growth of U.S. Population, 2000–2009

    Source: http://www.census.gov/popest/data.

    (Hint: To obtain the population in July 2001, multiply the population in July 2000 by 1.0112 to get 284,573,831.)

    Answer:
    309,389,957

     

    In the following exercises, estimate the areas under the curves by computing the left Riemann sums, \(\displaystyle L_8.\)

    42)

    43)

    Answer:
    \(\displaystyle L_8=3+2+1+2+3+4+5+4=24\)

    44)

    45)

    Answer:
    \(\displaystyle L_8=3+5+7+6+8+6+5+4=44\)

    46) [T] Use a computer algebra system to compute the Riemann sum, \(\displaystyle L_N\), for \(\displaystyle N=10,30,50\) for \(\displaystyle f(x)=\sqrt{1−x^2}\) on \(\displaystyle [−1,1].\)

    47) [T] Use a computer algebra system to compute the Riemann sum, \(\displaystyle L_N\), for \(\displaystyle N=10,30,50\) for \(\displaystyle f(x)=\frac{1}{\sqrt{1+x^2}}\) on \(\displaystyle [−1,1].\)

    Answer:
    \(\displaystyle L_{10}≈1.7604,L_{30}≈1.7625,L_{50}≈1.76265\)

    48) [T] Use a computer algebra system to compute the Riemann sum, \(\displaystyle L_N\), for \(\displaystyle N=10,30,50\) for \(\displaystyle f(x)=sin^2x\) on \(\displaystyle [0,2π]\). Compare these estimates with π.

     

    In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(\displaystyle R_N\) and \(\displaystyle L_N\) for \(\displaystyle N=1,10,100\). How do these estimates compare with the exact answers, which you can find via geometry?

    49) [T] \(\displaystyle y=cos(πx)\) on the interval \(\displaystyle [0,1]\)

    Answer:
    \(\displaystyle R_1=−1,L_1=1,R_{10}=−0.1,L_{10}=0.1,L_{100}=0.01,\) and \(\displaystyle R_{100}=−0.1.\) By symmetry of the graph, the exact area is zero.

    50) [T] \(\displaystyle y=3x+2\) on the interval \(\displaystyle [3,5]\)

     

    In the following exercises, use a calculator or a computer program to evaluate the endpoint sums \(\displaystyle R_N\) and \(\displaystyle L_N\) for \(\displaystyle N=1,10,100.\)

    51) [T] \(\displaystyle y=x^4−5x^2+4\) on the interval \(\displaystyle [−2,2]\), which has an exact area of \(\displaystyle \frac{32}{15}\)

    Answer:
    \(\displaystyle R_1=0,L_1=0,R_{10}=2.4499,L_{10}=2.4499,R_{100}=2.1365,L_{100}=2.1365\)

    52) [T] \(\displaystyle y=lnx\) on the interval \(\displaystyle [1,2]\), which has an exact area of \(\displaystyle 2ln(2)−1\)

    53) Explain why, if \(\displaystyle f(a)≥0\) and f is increasing on \(\displaystyle [a,b]\), that the left endpoint estimate is a lower bound for the area below the graph of \(\displaystyle f\) on \(\displaystyle [a,b].\)

    Answer:
    If \(\displaystyle [c,d]\)is a subinterval of \(\displaystyle [a,b]\) under one of the left-endpoint sum rectangles, then the area of the rectangle contributing to the left-endpoint estimate is \(\displaystyle f(c)(d−c)\). But, \(\displaystyle f(c)≤f(x)\) for \(\displaystyle c≤x≤d\), so the area under the graph of \(\displaystyle f\) between \(\displaystyle c\) and \(\displaystyle d\) is \(\displaystyle f(c)(d−c)\) plus the area below the graph of f but above the horizontal line segment at height \(\displaystyle f(c)\), which is positive. As this is true for each left-endpoint sum interval, it follows that the left Riemann sum is less than or equal to the area below the graph of \(\displaystyle f\) on \(\displaystyle [a,b].\)

    54) Explain why, if \(\displaystyle f(b)≥0\) and f is decreasing on \(\displaystyle [a,b],\) that the left endpoint estimate is an upper bound for the area below the graph of \(\displaystyle f\) on \(\displaystyle [a,b].\)

    55) Show that, in general, \(\displaystyle R_N−L_N=(b−a)×\frac{f(b)−f(a)}{N}.\)

    Answer:
    \(\displaystyle L_N=\frac{b−a}{N}\sum_{i=1}^Nf(a+(b−a)\frac{i−1}{N})=\frac{b−a}{N}\sum_{i=0}^{N−1}f(a+(b−a)\frac{i}{N})\) and \(\displaystyle R_N=\frac{b−a}{N}\sum_{i=1}^Nf(a+(b−a)\frac{i}{N})\). The left sum has a term corresponding to \(\displaystyle i=0\) and the right sum has a term corresponding to \(\displaystyle i=N\). In \(\displaystyle R_N−L_N\), any term corresponding to \(\displaystyle i=1,2,…,N−1\) occurs once with a plus sign and once with a minus sign, so each such term cancels and one is left with \(\displaystyle R_N−L_N=\frac{b−a}{N}(f(a+(b−a))\frac{N}{N})−(f(a)+(b−a)\frac{0}{N})=\frac{b−a}{N}(f(b)−f(a)).\)

    56) Explain why, if f is increasing on [ \(\displaystyle a,b\)], the error between either \(\displaystyle L_N\) or \(\displaystyle R_N\) and the area A below the graph of f is at most \(\displaystyle (b−a)\frac{f(b)−f(a)}{N]|).

    57) For each of the three graphs:

    a. Obtain a lower bound \(\displaystyle L(A)\) for the area enclosed by the curve by adding the areas of the squares enclosed completely by the curve.

    b. Obtain an upper bound \(\displaystyle U(A)\) for the area by adding to \(\displaystyle L(A)\) the areas \(\displaystyle B(A)\) of the squares enclosed partially by the curve.

     

    Answer:
    Graph 1: a. \(\displaystyle L(A)=0,B(A)=20; b. U(A)=20.\) Graph 2: \(\displaystyle a. L(A)=9; b. B(A)=11,U(A)=20.\) Graph 3: a. \(\displaystyle L(A)=11.0; b. B(A)=4.5,U(A)=15.5.\)

    58) In the previous exercise, explain why \(\displaystyle L(A)\) gets no smaller while \(\displaystyle U(A)\) gets no larger as the squares are subdivided into four boxes of equal area.

    59) A unit circle is made up of n wedges equivalent to the inner wedge in the figure. The base of the inner triangle is 1 unit and its height is \(\displaystyle sin(\frac{π}{n}).\) The base of the outer triangle is \(\displaystyle B=cos(\frac{π}{n})+sin(\frac{π}{n})tan(\frac{π}{n})\) and the height is \(\displaystyle H=Bsin(\frac{2π}{n})\). Use this information to argue that the area of a unit circle is equal to π.\)

    Answer:
    Let A be the area of the unit circle. The circle encloses n congruent triangles each of area \(\displaystyle \frac{sin(\frac{2π}{n})}{2}\), so \(\displaystyle \frac{n}{2}sin(\frac{2π}{n})≤A.\) Similarly, the circle is contained inside n congruent triangles each of area \(\displaystyle \frac{BH}{2}=\frac{1}{2}(cos(\frac{π}{n})+sin(\frac{π}{n})tan(\frac{π}{n}))sin(\frac{2π}{n})\), so \(\displaystyle A≤\frac{n}{2}sin(\frac{2π}{n})(cos(\frac{π}{n}))+sin(\frac{π}{n})tan(\frac{π}{n})\). As \(\displaystyle n→∞,\frac{n}{2}sin(\frac{2π}{n})=\frac{πsin(\frac{2π}{n})}{(\frac{2π}{n})}→π\), so we conclude \(\displaystyle π≤A\). Also, as \(\displaystyle n→∞,cos(\frac{π}{n})+sin(\frac{π}{n})tan(\frac{π}{n})→1\), so we also have \(\displaystyle A≤π\). By the squeeze theorem for limits, we conclude that \(\displaystyle A=π.\)

     

    J5.1.1) Consider the area under the curve  \(\displaystyle f(x) =ln(x)\) on \(\displaystyle [2,6]\).

    (a) If we use 20 subintervals, what is the length of each subinterval?

    (b) What condition does this function have on this interval that allows us to use the Maximum Possible Error formula?

    (c) What is the maximum possible error when a Riemann sum is used to estimate the exact value of the area under the curve with n = 8 for \(\displaystyle ln(x)\) on \(\displaystyle [2,6]\)?

    (d) How many subintervals do we need to use for the maximum possible error to be no more than 0.001?

     

    J5.1.2) Consider the area under the curve  \(\displaystyle g(x)=cos(πx)\) on \(\displaystyle [0,1]\).

    a) Find \(\displaystyle L_4\) for \(\displaystyle g(x)=cos(πx)\) on \(\displaystyle [0,1]\).

    b) Find \(\displaystyle R_4\) for \(\displaystyle g(x)=cos(πx)\) on \(\displaystyle [0,1]\).

    c) Look at a graph of \(\displaystyle g(x)=cos(πx)\) on \(\displaystyle [0,1]\).  What do you think the actual area for this region is?

    d) Do Riemann sums estimate net area or total area?