Chapter 6

For each region \(R\), find the horizontal line \(y=k\) that divides \(R\) into two subregions of equal area. \(R\) is the region bounded by \(y=1-|x-1|\) and the \(x\) -axis.

Definite integrals Evaluate the following definite integrals. Use Theorem \(6.10\) to express your answer in terms of logarithms. $$\int_{-2}^{2} \frac{d t}{t^{2}-9}$$

Let $$f(x)=\left\\{\begin{array}{cl}x & \text { if } 0 \leq x \leq 2 \\\2 x-2 & \text { if } 2

Suppose a cylindrical glass with a diameter of \(\frac{1}{12} \mathrm{m}\) and a height of \(\frac{1}{10} \mathrm{m}\) is filled to the brim with a 400-Cal milkshake. If you have a straw that is 1.1 m long (so the top of the straw is \(1 \mathrm{m}\) above the top of the glass), do you burn off all the calories in the milkshake in drinking it? Assume that the density of the milkshake is \(1 \mathrm{g} / \mathrm{cm}^{3}(1 \mathrm{Cal}=4184 \mathrm{J})\)

Definite integrals Evaluate the following definite integrals. Use Theorem \(6.10\) to express your answer in terms of logarithms. $$\int_{1 / 6}^{1 / 4} \frac{d t}{t \sqrt{1-4 t^{2}}}$$

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{\pi} 2^{\sin x} \cos x d x\)

Find the volume of the following solids using the method of your choice. The solid formed when the region bounded by \(y=\sqrt{x}\), the \(x\) -axis, and \(x=4\) is revolved about the \(x\) -axis

Sketch a solid of revolution whose volume by the disk method is given by the following integrals. Indicate the function that generates the solid. Solutions are not unique. a. \(\int_{0}^{\pi} \pi \sin ^{2} x d x\) b. \(\int_{0}^{2} \pi\left(x^{2}+2 x+1\right) d x\)

A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is \(2 \mathrm{m}\) on a side, and its lower edge is \(1 \mathrm{m}\) from the bottom of the tank. a. If the tank is filled to a depth of \(4 \mathrm{m},\) will the window withstand the resulting force? b. What is the maximum depth to which the tank can be filled without the window failing?

Definite integrals Evaluate the following definite integrals. Use Theorem \(6.10\) to express your answer in terms of logarithms. $$\int_{1 / 8}^{1} \frac{d x}{x \sqrt{1+x^{2 / 3}}}$$

Miscellaneous integrals Evaluate the following integrals. \(\int_{1}^{2 e} \frac{3^{\ln x}}{x} d x\)

Consider the region \(R\) bounded by the curves \(y=a x^{2}+1, y=0, x=0,\) and \(x=1,\) for \(a \geq-1 .\) Let \(S_{1}\) and \(S_{2}\) be solids generated when \(R\) is revolved about the \(x\) - and \(y\) -axes, respectively. a. Find \(V_{1}\) and \(V_{2}\), the volumes of \(S_{1}\) and \(S_{2}\), as functions of \(a\) b. What are the values of \(a \geq-1\) for which \(V_{1}(a)=V_{2}(a) ?\)

A typical human heart pumps \(70 \mathrm{mL}\) of blood with each stroke (stroke volume). Assuming a heart rate of 60 beats \(/ \min (1 \text { beat } / \mathrm{s}),\) a reasonable model for the outflow rate of the heart is \(V^{\prime}(t)=70(1+\sin 2 \pi t),\) where \(V(t)\) is the amount of blood (in milliliters) pumped over the interval \([0, t]\) \(V(0)=0,\) and \(t\) is measured in seconds. a. Graph the outflow rate function. b. Verify that the amount of blood pumped over a one-second interval is 70 mL. c. Find the function that gives the total blood pumped between \(t=0\) and a future time \(t>0\) d. What is the cardiac output over a period of 1 min? (Use calculus; then check your answer with algebra.)

Archimedes' principle says that the buoyant force exerted on an object that is (partially or totally) submerged in water is equal to the weight of the water displaced by the object (see figure). Let \(\rho_{w}=1 \mathrm{g} / \mathrm{cm}^{3}=1000 \mathrm{kg} / \mathrm{m}^{3}\) be the density of water and let \(\rho\) be the density of an object in water. Let \(f=\rho / \rho_{w}\). If \(01,\) then the object sinks. Consider a cubical box with sides \(2 \mathrm{m}\) long floating in water with one-half of its volume submerged \(\left(\rho=\rho_{w} / 2\right) .\) Find the force required to fully submerge the box (so its top surface is at the water level).

Definite integrals Evaluate the following definite integrals. Use Theorem \(6.10\) to express your answer in terms of logarithms. $$\int_{\ln 5}^{\ln 9} \frac{\cosh x}{4-\sinh ^{2} x} d x$$

Miscellaneous integrals Evaluate the following integrals. \(\int \frac{\sin (\ln x)}{4 x} d x\)

Let \(R\) be the region in the first quadrant bounded by the circle \(x^{2}+y^{2}=r^{2}\) and the coordinate axes. Find the volume of a hemisphere of radius \(r\) in the following ways. a. Revolve \(R\) about the \(x\) -axis and use the disk method. b. Revolve \(R\) about the \(x\) -axis and use the shell method. c. Assume the base of the hemisphere is in the \(x y\) -plane and use the general slicing method with slices perpendicular to the \(x y\) -plane and parallel to the \(x\) -axis.

A right circular cylinder with height \(R\) and radius \(R\) has a volume of \(V_{C}=\pi R^{3}\) (height \(=\) radius). a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height \(R\). Express the volume in terms of \(V_{C}\). b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of \(V_{C}\).

A simple model (with different parameters for different people) for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi}{2} \sin \frac{\pi t}{2}$$ where \(V(t)\) (measured in liters) is the volume of air in the lungs at time \(t \geq 0, t\) is measured in seconds, and \(t=0\) corresponds to a time at which the lungs are full and exhalation begins. Only a fraction of the air in the lungs in exchanged with each breath. The amount that is exchanged is called the tidal volume. a. Find and graph the volume function \(V\) assuming that $$ V(0)=6 \mathrm{L} $$ b. What is the breathing rate in breaths/min? c. What is the tidal volume and what is the total capacity of the lungs?

Miscellaneous integrals Evaluate the following integrals. \(\int_{1}^{e^{2}} \frac{(\ln x)^{5}}{x} d x\)

Verify that the volume of a right circular cone with a base radius of \(r\) and a height of \(h\) is \(\pi r^{2} h / 3 .\) Use the region bounded by the line \(y=r x / h,\) the \(x\) -axis, and the line \(x=h,\) where the region is rotated about the \(x\) -axis. Then (a) use the disk method and integrate with respect to \(x,\) and (b) use the shell method and integrate with respect to \(y\)

A hemispherical bowl of radius 8 inches is filled to a depth of \(h\) inches, where \(0 \leq h \leq 8\). Find the volume of water in the bowl as a function of \(h\). (Check the special cases \(h=0 \text { and } h=8 .)\)

Some species have growth rates that oscillate with an (approximately) constant period \(P\). Consider the growth rate function $$N^{\prime}(t)=r+A \sin \frac{2 \pi t}{P}$$ where \(A\) and \(r\) are constants with units of individuals/yr, and \(t\) is measured in years. A species becomes extinct if its population ever reaches 0 after \(t=0\) a. Suppose \(P=10, A=20,\) and \(r=0 .\) If the initial population is \(N(0)=10,\) does the population ever become extinct? Explain. b. Suppose \(P=10, A=20,\) and \(r=0 .\) If the initial population is \(N(0)=100,\) does the population ever become extinct? Explain. c. Suppose \(P=10, A=50,\) and \(r=5 .\) If the initial population is \(N(0)=10,\) does the population ever become extinct? Explain. d. Suppose \(P=10, A=50,\) and \(r=-5 .\) Find the initial population \(N(0)\) needed to ensure that the population never becomes extinct.

Consider the parabola \(y=x^{2} .\) Let \(P, Q,\) and \(R\) be points on the parabola with \(R\) between \(P\) and \(Q\) on the curve. Let \(\ell_{p}, \ell_{Q},\) and \(\ell_{R}\) be the lines tangent to the parabola at \(P, Q,\) and \(R,\) respectively (see figure). Let \(P^{\prime}\) be the intersection point of \(\ell_{Q}\) and \(\ell_{R},\) let \(Q^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{R},\) and let \(R^{\prime}\) be the intersection point of \(\ell_{P}\) and \(\ell_{Q} .\) Prove that Area \(\Delta P Q R=2 \cdot\) Area \(\Delta P^{\prime} Q^{\prime} R^{\prime}\) in the following cases. (In fact, the property holds for any three points on any parabola.) (Source: Mathematics Magazine 81, 2, Apr 2008) a. \(P\left(-a, a^{2}\right), Q\left(a, a^{2}\right),\) and \(R(0,0),\) where \(a\) is a positive real number b. \(P\left(-a, a^{2}\right), Q\left(b, b^{2}\right),\) and \(R(0,0),\) where \(a\) and \(b\) are positive real numbers c. \(P\left(-a, a^{2}\right), Q\left(b, b^{2}\right),\) and \(R\) is any point between \(P\) and \(Q\) on the curve

Show that the arc length of the catenary \(y=\cosh x\) over the interval \([0, a]\) is \(L=\sinh\) \(a.\)

Miscellaneous integrals Evaluate the following integrals. \(\int \frac{\ln ^{2} x+2 \ln x-1}{x} d x\)

Find the volume of the torus formed when the circle of radius 2 centered at (3,0) is revolved about the \(y\) -axis. Use geometry to evaluate the integral.

Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal = 4184 J. One hour of walking consumes roughly \(10^{6} \mathrm{J},\) or \(250 \mathrm{Cal} .\) On the other hand, power is the rate at which energy is used and is measured in watts \((\mathrm{W} ; 1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})\) Other useful units of power are kilowatts ( \(1 \mathrm{kW}=10^{3} \mathrm{W}\) ) and megawatts ( \(1 \mathrm{MW}=10^{6} \mathrm{W}\) ). If energy is used at a rate of \(1 \mathrm{kW}\) for 1 hr, the total amount of energy used is 1 kilowatt-hour \((\mathrm{kWh})=\) which is \(3.6 \times 10^{6} \mathrm{J}\) Suppose the power function of a large city over a \(24-\mathrm{hr}\) period is given by $$P(t)=E^{\prime}(t)=300-200 \sin \frac{\pi t}{12}$$ where \(P\) is measured in megawatts and \(t=0\) corresponds to 6:00 P.M. (see figure). a. How much energy is consumed by this city in a typical \(24-\mathrm{hr}\) period? Express the answer in megawatt-hours and in joules. b. Burning 1 kg of coal produces about 450 kWh of energy. How many kg of coal are required to meet the energy needs of the city for 1 day? For 1 year? c. Fission of 1 g of uranium- 235 (U-235) produces about \(16,000 \mathrm{kWh}\) of energy. How many grams of uranium are needed to meet the energy needs of the city for 1 day? For 1 year? d. A typical wind turbine can generate electrical power at a rate of about \(200 \mathrm{kW}\). Approximately how many wind turbines are needed to meet the average energy needs of the city?

A power line is attached at the same height to two utility poles that are separated by a distance of \(100 \mathrm{ft}\); the power line follows the curve \(f(x)=a \cosh (x / a) .\) Use the following steps to find the value of \(a\) that produces a sag of \(10 \mathrm{ft}\) midway between the poles. Use a coordinate system that places the poles at \(x=\pm 50.\) a. Show that \(a\) satisfies the equation \(\cosh (50 / a)-1=10 / a.\) b. Let \(t=10 / a,\) confirm that the equation in part (a) reduces to \(\cosh 5 t-1=t,\) and solve for \(t\) using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find \(a\) and then compute the length of the power line.

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{\ln 2} \frac{e^{3 x}-e^{-3 x}}{e^{3 x}+e^{-3 x}} d x\)

A hemispherical bowl of radius 8 inches is filled to a depth of \(h\) inches, where \(0 \leq h \leq 8(h=0\) corresponds to an empty bowl). Use the shell method to find the volume of water in the bowl as a function of \(h\). (Check the special cases \(h=0\) and \(h=8 .)\)

At Earth's surface, the acceleration due to gravity is approximately \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) (with local variations). However, the acceleration decreases with distance from the surface according to Newton's law of gravitation. At a distance of \(y\) meters from Earth's surface, the acceleration is given by where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of Earth. a. Suppose a projectile is launched upward with an initial velocity of \(v_{0} \mathrm{m} / \mathrm{s} .\) Let \(v(t)\) be its velocity and \(y(t)\) its height (in meters) above the surface \(t\) seconds after the launch. Neglecting forces such as air resistance, explain why \(\frac{d v}{d t}=a(y)\) and \(\frac{d y}{d t}=v(t)\) b. Use the Chain Rule to show that \(\frac{d v}{d t}=\frac{1}{2} \frac{d}{d y}\left(v^{2}\right)\) c. Show that the equation of motion for the projectile is \(\frac{1}{2} \frac{d}{d y}\left(v^{2}\right)=a(y),\) where \(a(y)\) is given previously. d. Integrate both sides of the equation in part (c) with respect to \(y\) using the fact that when \(y=0, v=v_{0} .\) Show that $$ \frac{1}{2}\left(v^{2}-v_{0}^{2}\right)=g R\left(\frac{1}{1+y / R}-1\right) $$ e. When the projectile reaches its maximum height, \(v=0\) Use this fact to determine that the maximum height is \(y_{\max }=\frac{R v_{0}^{2}}{2 g R-v_{0}^{2}}\) f. Graph \(y_{\max }\) as a function of \(v_{0} .\) What is the maximum height when \(v_{0}=500 \mathrm{m} / \mathrm{s}, 1500 \mathrm{m} / \mathrm{s},\) and \(5 \mathrm{km} / \mathrm{s} ?\) g. Show that the value of \(v_{0}\) needed to put the projectile into orbit (called the escape velocity) is \(\sqrt{2 g R}\)

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{1} \frac{16^{x}}{4^{2 x}} d x\)

Suppose that \(f\) and \(g\) have continuous derivatives on an interval \([a, b] .\) Prove that if \(f(a)=g(a)\) and \(f(b)=g(b),\) then \(\int_{a}^{b} f^{\prime}(x) d x=\int_{a}^{b} g^{\prime}(x) d x\)

Two points \(P\) and \(Q\) are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment \(P Q\) is less than one-fourth the area of the square? Begin by showing that \(x\) and \(y\) must satisfy \(x y<\frac{1}{2}\) in order for the area condition to be met. Then argue that the required probability is \(\frac{1}{2}+\int_{1 / 2}^{1} \frac{d x}{2 x}\) and evaluate the integral.

Find the volume of the torus formed when a circle of radius 2 centered at (3,0) is revolved about the \(y\) -axis. Use the shell method. You may need a computer algebra system or table of integrals to evaluate the integral.

Consider the region \(R\) in the first quadrant bounded by \(y=x^{1 / n}\) and \(y=x^{n},\) where \(n>1\) is a positive number. a. Find the volume \(V(n)\) of the solid generated when \(R\) is revolved about the \(x\) -axis. Express your answer in terms of \(n\). b. Evaluate \(\lim _{n \rightarrow \infty} V(n) .\) Interpret this limit geometrically.

Find the area of the region bounded by the curve \(x=\frac{1}{2 y}-\sqrt{\frac{1}{4 y^{2}}-1}\) and the line \(x=1\) in the first quadrant. (Hint: Express \(y\) in terms of \(x\).)

Differentiate \(\ln x\) for \(x>0\) and differentiate \(\ln (-x)\) for \(x<0\) to conclude that \(\frac{d}{d x}(\ln |x|)=\frac{1}{x}\).

Suppose \(R\) is the region bounded by \(y=f(x)\) and \(y=g(x)\) on the interval \([a, b],\) where \(f(x) \geq g(x)\) a. Show that if \(R\) is revolved about the vertical line \(x=x_{0}\) where \(x_{0}b ?\)

Use Exercise 69 to prove that if two trails start at the same place and finish at the same place, then regardless of the ups and downs of the trails, they have the same net change in elevation.

Consider the cubic polynomial \(f(x)=x(x-a)(x-b),\) where \(0 \leq a \leq b\) a. For a fixed value of \(b,\) find the function \(F(a)=\int_{0}^{b} f(x) d x\) For what value of \(a\) (which depends on \(b\) ) is \(F(a)=0 ?\) b. For a fixed value of \(b\), find the function \(A(a)\) that gives the area of the region bounded by the graph of \(f\) and the \(x\) -axis between \(x=0\) and \(x=b\). Graph this function and show that it has a minimum at \(a=b / 2\). What is the maximum value of \(A(a),\) and where does it occur (in terms of \(b\) )?

a. Confirm that the linear approximation to \(f(x)=\tanh x\) at \(a=0\) is \(L(x)=x.\) b. Recall that the velocity of a surface wave on the ocean is \(v=\sqrt{\frac{g \lambda}{2 \pi} \tanh \frac{2 \pi d}{\lambda}} .\) In fluid dynamics, shallow water refers to water where the depth-to-wavelength ratio \(d / \lambda<0.05 .\) Use your answer to part (a) to explain why the shallow water velocity equation is \(v=\sqrt{g d}.\) c. Use the shallow-water velocity equation to explain why waves tend to slow down as they approach the shore.

Use the inverse relations between \(\ln x\) and \(e^{x}(\exp (x)),\) and the properties of \(\ln x\) to prove the following properties. a. \(\exp (0)=1\) b. \(\exp (x-y)=\frac{\exp (x)}{\exp (y)}\) c. \((\exp (x))^{p}=\exp (p x), p\) rational

Suppose \(R\) is the region bounded by \(y=f(x)\) and \(y=g(x)\) on the interval \([a, b],\) where \(f(x) \geq g(x) \geq 0\) a. Show that if \(R\) is revolved about the horizontal line \(y=y_{0}\) that lies below \(R,\) then by the washer method, the volume of the resulting solid is $$ V=\int_{a}^{b} \pi\left(\left(f(x)-y_{0}\right)^{2}-\left(g(x)-y_{0}\right)^{2}\right) d x $$ b. How is this formula changed if the line \(y=y_{0}\) lies above \(R ?\)

Without evaluating integrals, prove that $$ \int_{0}^{2} \frac{d}{d x}\left(12 \sin \pi x^{2}\right) d x=\int_{0}^{2} \frac{d}{d x}\left(x^{10}(2-x)^{3}\right) d x $$

Assume \(f\) and \(g\) are even, integrable functions on \([-a, a],\) where \(a>1 .\) Suppose \(f(x)>g(x)>0\) on \([-a, a]\) and the area bounded by the graphs of \(f\) and \(g\) on \([-a, a]\) is \(10 .\) What is the value of \(\int_{0}^{\sqrt{a}} x\left(f\left(x^{2}\right)-g\left(x^{2}\right)\right) d x ?\)

A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging between \(150\) to \(1000 \mathrm{km} .\) Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about \(4000 \mathrm{m}.\) Explain why the shallow-water velocity equation (Exercise 71 ) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use \(d=4000 \mathrm{m}\) )?

Use the following argument to show that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\). a. Make a sketch of the function \(f(x)=1 / x\) on the interval \([1,2] .\) Explain why the area of the region bounded by \(y=f(x)\) and the \(x\) -axis on [1,2] is \(\ln 2\) b. Construct a rectangle over the interval [1,2] with height \(\frac{1}{2}\) Explain why \(\ln 2>\frac{1}{2}\). c. Show that \(\ln 2^{n}>n / 2\) and \(\ln 2^{-n}<-n / 2\). d. Conclude that \(\lim _{x \rightarrow \infty} \ln x=\infty\) and \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\).

An ellipse centered at the origin is described by the equation \(x^{2} / a^{2}+y^{2} / b^{2}=1 .\) If an ellipse \(R\) is revolved about either axis, the resulting solid is an ellipsoid. a. Find the volume of the ellipsoid generated when \(R\) is revolved about the \(x\) -axis (in terms of \(a\) and \(b\) ). b. Find the volume of the ellipsoid generated when \(R\) is revolved about the \(y\) -axis (in terms of \(a\) and \(b\) ). c. Should the results of parts (a) and (b) agree? Explain.