1.E: Chapter Review Exercises

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Chapter Review Exercises

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

Exercise $\PageIndex{1}$

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.

Answer: False

Exercise $\PageIndex{2}$

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{3}$

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.

Answer: False

Exercise $\PageIndex{4}$

If the half-life of $\displaystyle seaborgium-266$ is $\displaystyle 360$ ms, then $\displaystyle k=(ln(2))/360.$

Answer: Add texts here. Do not delete this text first.

For the following exercises, use the requested method to determine the volume of the solid.

Exercise $\PageIndex{5}$

The volume that has a base of the ellipse $\displaystyle x^2/4+y^2/9=1$ and cross-sections of an equilateral triangle perpendicular to the $\displaystyle y-axis$. Use the method of slicing.

Answer: $\displaystyle 32\sqrt{3}$

Exercise $\PageIndex{6}$

$\displaystyle y=x^2−x$, from $\displaystyle x=1$ to $\displaystyle x=4$, rotated around they-axis using the washer method

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{7}$

$\displaystyle x=y^2$ and $\displaystyle x=3y$ rotated around the y-axis using the washer method

Answer: $\displaystyle \frac{162π}{5}$

Exercise $\PageIndex{8}$

$\displaystyle x=2y^2−y^3,x=0$,and $\displaystyle y=0$ rotated around the x-axis using cylindrical shells

Answer: Add texts here. Do not delete this text first.

For the following exercises, 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.

Exercise $\PageIndex{9}$

$\displaystyle y=x^3,x=0,y=0$, and $\displaystyle x=2$

Answer: $\displaystyle a. 4, b. \frac{128π}{7}, c. \frac{64π}{5}$

Exercise $\PageIndex{10}$

$\displaystyle y=x^2−x$ and $\displaystyle x=0$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{11}$

$\displaystyle y=ln(x)+2$ and $\displaystyle y=x$

Answer: $\displaystyle a. 1.949, b. 21.952, c. 17.099$

Exercise $\PageIndex{12}$

$\displaystyle y=x^2$ and $\displaystyle y=\sqrt{x}$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{13}$

$\displaystyle y=5+x, y=x^2, x=0$, and $\displaystyle x=1$

Answer: $\displaystyle a. \frac{31}{6},b. \frac{452π}{15}, c. \frac{31π}{6}$

Exercise $\PageIndex{14}$

Below $\displaystyle x^2+y^2=1$ and above $\displaystyle y=1−x$

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{15}$

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

Answer: $\displaystyle 245.282$

Exercise $\PageIndex{16}$

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{17}$

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

Answer: Mass: $\displaystyle \frac{1}{2},$ center of mass: $\displaystyle (\frac{18}{35},\frac{9}{11})$

For the following exercises, find the requested arc lengths.

Exercise $\PageIndex{18}$

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{19}$

The length of $\displaystyle y$ for $\displaystyle x=3−\sqrt{y}$ from $\displaystyle y=0$ to $\displaystyle y=4$

Answer: $\displaystyle \sqrt{17}+\frac{1}{8}ln(33+8\sqrt{17})$

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

Exercise $\PageIndex{20}$

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

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{21}$

The loudspeaker created by revolving $\displaystyle y=1/x$ from $\displaystyle x=1$ to $\displaystyle x=4$ around the x-axis.

Answer: Volume: $\displaystyle \frac{3π}{4},$ surface area: $\displaystyle π(\sqrt{2}−sinh^{−1}(1)+sinh^{−1}(16)−\frac{\sqrt{257}}{16})$

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

Exercise $\PageIndex{22}$

Find the total force on the wall of the dam.

Answer: Add texts here. Do not delete this text first.

Exercise $\PageIndex{23}$

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

Answer: 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.

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

Source: http:/stockcharts.com/freecharts/hi...a19201940.html

Exercise $\PageIndex{24}$

The best-fit exponential curve to these data is given by $\displaystyle y=40.71+1.224^x$. 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?

Answer: Add texts here. Do not delete this text first.

For the following exercises, 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.

Exercise $\PageIndex{25}$

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

$alt$

Answer: $\displaystyle π(1+sinh(1)cosh(1))$

Exercise $\PageIndex{26}$

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

Answer: Add texts here. Do not delete this text first.