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Chapter 6 Review Exercises

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    True or False? Justify your answer with a proof or a counterexample.

    1) The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.


    2) If the force is constant, the amount of work to move an object from \(x=a\) to \(x=b\) is \(F(b−a)\).

    3) The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.


    4) If the half-life of \(seaborgium-266\) is \(360\) ms, then \(k=\dfrac{\ln 2}{360}.\)

    For exercises 5 - 8, use the requested method to determine the volume of the solid.

    5) The volume that has a base of the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\) and cross-sections of an equilateral triangle perpendicular to the \(y\)-axis. Use the method of slicing.

    \(V = 32\sqrt{3}\, \text{units}^3\)

    6) \(y=x^2−x\), from \(x=1\) to \(x=4\), rotated around the \(y\)-axis using the washer method

    7) \(x=y^2\) and \(x=3y\) rotated around the \(y\)-axis using the washer method

    \(V = \frac{162π}{5}\, \text{units}^3\)

    8) \(x=2y^2−y^3,\; x=0\),and \(y=0\) rotated around the \(x\)-axis using cylindrical shells

    For exercises 9 - 14, find

    a. the area of the region,

    b.the volume of the solid when rotated around the \(x\)-axis, and

    c. the volume of the solid when rotated around the \(y\)-axis. Use whichever method seems most appropriate to you.

    9) \(y=x^3,x=0,y=0\), and \(x=2\)

    a. \(A = 4\) units2
    b. \(V = \frac{128π}{7}\) units3
    c. \(V = \frac{64π}{5}\) units3

    10) \(y=x^2−x\) and \(x=0\)

    11) [T] \(y=\ln(x)+2\) and \(y=x\)

    a. \(A \approx 1.949\) units2
    b. \(V \approx 21.952\) units3
    c. \(V = \approx 17.099\) units3

    12) \(y=x^2\) and \(y=\sqrt{x}\)

    13) \(y=5+x, y=x^2, x=0\), and \(x=1\)

    a. \(A = \frac{31}{6}\) units2
    b. \(V = \frac{452π}{15}\) units3
    c. \(V = \frac{31π}{6}\) units3

    14) Below \(x^2+y^2=1\) and above \(y=1−x\)

    15) Find the mass of \(ρ=e^{−x}\) on a disk centered at the origin with radius \(4\).

    \(m \approx 245.282\)

    16) Find the center of mass for \(ρ=\tan^2x\) on \(x\in (−\frac{π}{4},\frac{π}{4})\).

    17) Find the mass and the center of mass of \(ρ=1\) on the region bounded by \(y=x^5\) and \(y=\sqrt{x}\).

    Mass: \(\frac{1}{2},\)
    Center of mass: \((\frac{18}{35},\frac{9}{11})\)

    For exercises 18 - 19, find the requested arc lengths.

    18) The length of \(x\) for \(y=\cosh(x)\) from \(x=0\) to \(x=2\).

    19) The length of \(y\) for \(x=3−\sqrt{y}\) from \(y=0\) to \(y=4\)

    \(s = \big[\sqrt{17}+\frac{1}{8}\ln(33+8\sqrt{17})\big]\) units

    For exercises 20 - 21, find the surface area and volume when the given curves are revolved around the specified axis.

    20) The shape created by revolving the region between \(y=4+x, \;y=3−x, \;x=0,\) and \(x=2\) rotated around the \(y\)-axis.

    21) The loudspeaker created by revolving \(y=\dfrac{1}{x}\) from \(x=1\) to \(x=4\) around the \(x\)-axis.

    Volume: \(V = \frac{3π}{4}\) units3
    Surface area: \(A = π\left(\sqrt{2}−\sinh^{−1}(1)+\sinh^{−1}(16)−\frac{\sqrt{257}}{16}\right)\) units2

    For exercise 22, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 m and width 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m3.

    22) Find the total force on the wall of the dam.

    23) You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and \(45\) °F outside and the temperature of the body is \(78\) °F. You know the cooling constant is \(k=0.00824\) °F/min. When did the victim die, assuming that a human’s temperature is \(98\) °F?

    11:02 a.m.

    For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash.

    Year after 1920 Value ($)
    1 63.90
    3 100
    5 110
    7 160
    9 381.17

    Source: http:/

    24) [T] The best-fit exponential curve to these data is given by \(y=40.71+1.224^x\). Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014 ?

    For exercises 25 - 26, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.

    25) Find the volume of the catenoid \(y=\cosh(x)\) from \(x=−1\) to \(x=1\) that is created by rotating this curve around the \(x\)-axis, as shown here.

    This figure is an image of a catenoid. It has been formed by rotating a catenary curve about a vertical axis.

    \(V = π\big(1+\sinh(1)\cosh(1)\big)\) units3

    26) Find surface area of the catenoid \(y=\cosh(x)\) from \(x=−1\) to \(x=1\) that is created by rotating this curve around the \(x\)-axis.


    • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at

    This page titled Chapter 6 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.