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Mathematics LibreTexts

4.6 E Exercises

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    13800
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    Use basic integration formulas to compute the following antiderivatives.

    1) \(\displaystyle ∫(\sqrt{x}−\frac{1}{\sqrt{x}})dx\)

    Solution: \(\displaystyle ∫(\sqrt{x}−\frac{1}{\sqrt{x}})dx=∫x^{1/2}dx−∫x^{−1/2}dx=\frac{2}{3}x^{3/2}+C_1−2x^{1/2+}C_2=\frac{2}{3}x^{3/2}−2x^{1/2}+C\)

    2) \(\displaystyle ∫(e^{2x}−\frac{1}{2}e^{x/2})dx\)

    3) \(\displaystyle ∫\frac{dx}{2x}\)

    Solution: \(\displaystyle ∫\frac{dx}{2x}=\frac{1}{2}ln|x|+C\)

    4) \(\displaystyle ∫\frac{x−1}{x^2}dx\)

    5) \(\displaystyle ∫^π_0(sinx−cosx)dx\)

    Solution: \(\displaystyle ∫^π_0sinxdx−∫^π_0cosxdx=−cosx|^π_0−(sinx)|^π_0=(−(−1)+1)−(0−0)=2\)

    6) \(\displaystyle ∫^{π/2}_0(x−sinx)dx\)

    7) Write an integral that expresses the increase in the perimeter \(\displaystyle P(s)\) of a square when its side length s increases from 2 units to 4 units and evaluate the integral.

    Solution: \(\displaystyle P(s)=4s,\) so \(\displaystyle \frac{dP}{ds}=4\) and \(\displaystyle ∫^4_24ds=8.\)

    8) Write an integral that quantifies the change in the area \(\displaystyle A(s)=s^2\)of a square when the side length doubles from S units to 2S units and evaluate the integral.

    9) A regular N-gon (an N-sided polygon with sides that have equal length s, such as a pentagon or hexagon) has perimeter Ns. Write an integral that expresses the increase in perimeter of a regular N-gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

    Solution: \(\displaystyle ∫^2_1Nds=N\)

    10) The area of a regular pentagon with side length \(\displaystyle a>0\) is \(\displaystyle pa^2\) with \(\displaystyle p=\frac{1}{4}\sqrt{5+\sqrt{5+2\sqrt{5}}}\). The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon according to these dimensions and evaluate this area.

    11) A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. By how much does the surface area of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?

    Solution: With p as in the previous exercise, each of the 12 pentagons increases in area from 2p to 4p units so the net increase in the area of the dodecahedron is 36punits.

    12) An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2a units?

    13) Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to 2s units and evaluate the integral.

    Solution: \(\displaystyle 18s^2=6∫^{2s}_s2xdx\)

    14) Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2s units and evaluate the integral.

    15) Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.

    Solution: \(\displaystyle 12πR^2=8π∫^{2R}_Rrdr\)

    16) Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.

    17) Suppose that a particle moves along a straight line with velocity \(\displaystyle v(t)=4−2t,\) where \(\displaystyle 0≤t≤2\) (in meters per second). Find the displacement at time t and the total distance traveled up to \(\displaystyle t=2.\)

    Solution: \(\displaystyle d(t)=∫^t_0v(s)ds=4t−t^2\). The total distance is \(\displaystyle d(2)=4m.\)

    18) Suppose that a particle moves along a straight line with velocity defined by \(\displaystyle v(t)=t^2−3t−18,\) where \(\displaystyle 0≤t≤6\) (in meters per second). Find the displacement at time t and the total distance traveled up to \(\displaystyle t=6.\)

    19) Suppose that a particle moves along a straight line with velocity defined by \(\displaystyle v(t)=|2t−6|,\) where \(\displaystyle 0≤t≤6\) (in meters per second). Find the displacement at time t and the total distance traveled up to \(\displaystyle t=6.\)

    Solution: \(\displaystyle d(t)=∫^t_0v(s)ds.\) For \(\displaystyle t<3,d(t)=∫^t_0(6−2t)dt=6t−t^2\). For \(\displaystyle t>3,d(t)=d(3)+∫^t_3(2t−6)dt=9+(t^2−6t)\). The total distance is \(\displaystyle d(6)=9m.\)

    20) Suppose that a particle moves along a straight line with acceleration defined by \(\displaystyle a(t)=t−3,\) where \(\displaystyle 0≤t≤6\) (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to \(\displaystyle t=6\) if \(\displaystyle v(0)=3\) and \(\displaystyle d(0)=0.\)

    21) A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec2. Neglecting air resistance, solve for the velocity \(\displaystyle v(t)\) and the height \(\displaystyle h(t)\) of the ball t seconds after it is thrown and before it returns to the ground.

    Solution: \(\displaystyle v(t)=40−9.8t;h(t)=1.5+40t−4.9t^2\)m/s

    22) A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is \(\displaystyle −9.8 m/sec^2\). Neglecting air resistance, solve for the velocity \(\displaystyle v(t)\) and the height \(\displaystyle h(t)\) of the ball t seconds after it is thrown and before it returns to the ground.

    23) The area \(\displaystyle A(t)\) of a circular shape is growing at a constant rate. If the area increases from 4π units to 9π units between times \(\displaystyle t=2\) and \(\displaystyle t=3,\) find the net change in the radius during that time.

    Solution: The net increase is 1 unit.

    24) A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from \(\displaystyle 36π in.^3\) to \(\displaystyle 288π in.^3\) between time \(\displaystyle t=30\) and \(\displaystyle t=60\) seconds, find the net change in the radius of the balloon during that time.

    25) Water flows into a conical tank with cross-sectional area \(\displaystyle πx^2\) at height x and volume \(\displaystyle \frac{πx^3}{3}\) up to height x. If water flows into the tank at a rate of 1 \(\displaystyle m^3/min\), find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.

    Solution: At \(\displaystyle t=5\), the height of water is \(\displaystyle x=(\frac{15}{π})^{1/3}m..\) The net change in height from \(\displaystyle t=5\) to \(\displaystyle t=10\) is \(\displaystyle (\frac{(30}{π})^{1/3}−(\frac{15}{π})^{1/3}m.\)

    26) A horizontal cylindrical tank has cross-sectional area \(\displaystyle A(x)=4(6x−x^2)m^2\) at height x meters above the bottom when \(\displaystyle x≤3.\)

    a. The volume V between heights a and b is \(\displaystyle ∫^b_aA(x)dx.\) Find the volume at heights between 2 m and 3 m.

    b. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, \(\displaystyle \frac{dx}{dt}=\frac{dx}{dV}\frac{dV}{dt}\), at how many meters per minute is the height of oil in the tank changing, expressed in terms of x, when the height is at x meters?

    c. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?

    27) The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical 24-hour period, with hour 1 corresponding to midnight to 1 a.m.

    Hour Power Hour Power
    1 28 13 48
    2 25 14 49
    3 24 15 49
    4 23 16 50
    5 24 17 50
    6 27 18 50
    7 29 19 46
    8 32 20 43
    9 34 21 42
    10 39 22 40
    11 42 23 37
    12 46 24 34

    Find the total amount of power in gigawatt-hours (gW-h) consumed by the city in a typical 24-hour period.

    Solution: The total daily power consumption is estimated as the sum of the hourly power rates, or 911 gW-h.

    28) The average residential electrical power use (in hundreds of watts) per hour is given in the following table.

    Hour Power Hour Power
    1 8 13 12
    2 6 14 13
    3 5 15 14
    4 4 16 15
    5 5 17 17
    6 6 18 19
    7 7 19 18
    8 8 20 17
    9 9 21 16
    10 10 22 16
    11 10 23 13
    12 11 24 11

    a. Compute the average total energy used in a day in kilowatt-hours (kWh).

    b. If a ton of coal generates 1842 kWh, how long does it take for an average residence to burn a ton of coal?

    c. Explain why the data might fit a plot of the form \(\displaystyle p(t)=11.5−7.5sin(\frac{πt}{12}).\)

    29) The data in the following table are used to estimate the average power output produced by Peter Sagan for each of the last 18 sec of Stage 1 of the 2012 Tour de France.

    Second Watts Second Watts
    1 600 10 1200
    2 500 11 1170
    3 575 12 1125
    4 1050 13 1100
    5 925 14 1075
    6 950 15 1000
    7 1050 16 950
    8 950 17 900
    9 1100 18 780

    Average Power OutputSource: sportsexercisengineering.com

    Estimate the net energy used in kilojoules (kJ), noting that 1W = 1 J/s, and the average power output by Sagan during this time interval.

    Solution: 17 kJ

    30) The data in the following table are used to estimate the average power output produced by Peter Sagan for each 15-min interval of Stage 1 of the 2012 Tour de France.

    Minutes Watts Minutes Watts
    15 200 165 170
    30 180 180 220
    45 190 195 140
    60 230 210 225
    75 240 225 170
    90 210 240 10
    105 210 255 200
    1120 220 270 220
    135 210 285 250
    150 150 300 400

    Average Power OutputSource: sportsexercisengineering.com

    Estimate the net energy used in kilojoules, noting that 1W = 1 J/s.

    31) The distribution of incomes as of 2012 in the United States in $5000 increments is given in the following table. The kth row denotes the percentage of households with incomes between \(\displaystyle $5000xk\) and \(\displaystyle 5000xk+4999\). The row \(\displaystyle k=40\) contains all households with income between $200,000 and $250,000 and \(\displaystyle k=41\) accounts for all households with income exceeding $250,000.

    0 3.5 21 1.5
    1 4.1 22 1.4
    2 5.9 23 1.3
    3 5.7 24 1.3
    4 5.9 25 1.1
    5 5.4 26 1.0
    6 5.5 27 0.75
    7 5.1 28 0.8
    8 4.8 29 1.0
    9 4.1 30 0.6
    10 4.3 31 0.6
    11 3.5 32 .5
    12 3.7 33 0.5
    13 3.2 34 0.4
    14 3.0 35 0.3
    15 2.8 36 0.3
    16 2.5 37 0.3
    17 2.2 38 0.2
    18 2.2 39 1.8
    19 1.8 40 2.3
    20 2.1 41  

    Income DistributionsSource: http://www.census.gov/prod/2013pubs/p60-245.pdf

    a. Estimate the percentage of U.S. households in 2012 with incomes less than $55,000.

    b. What percentage of households had incomes exceeding $85,000?

    c. Plot the data and try to fit its shape to that of a graph of the form \(\displaystyle a(x+c)e^{−b(x+e)}\)  for suitable \(\displaystyle a,b,c.\)

    Solution: \(\displaystyle a. 54.3%; b. 27.00%; c. \)The curve in the following plot is \(\displaystyle 2.35(t+3)e^{−0.15(t+3)}.\)