Chapter 6

Determine the following indefinite integrals. \(\int \frac{d x}{8-x^{2}}, x>2 \sqrt{2}\)

Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. $$v(t)=t\left(25-t^{2}\right)^{1 / 2}, \text { for } 0 \leq t \leq 5$$

Let \(R\) be the region bounded by the curve \(y=\sqrt{x+a}\) (with \(a > 0\) ), the \(y\) -axis, and the \(x\) -axis. Let \(S\) be the solid generated by rotating \(R\) about the \(y\) -axis. Let \(T\) be the inscribed cone that has the same circular base as \(S\) and height \(\sqrt{a} .\) Show that volume \((S) /\) volume \((T)=\frac{8}{5}\)

A rigid body with a mass of \(2 \mathrm{kg}\) moves along a line due to a force that produces a position function \(x(t)=4 t^{2},\) where \(x\) is measured in meters and \(t\) is measured in seconds. Find the work done during the first 5 s in two ways. a. Note that \(x^{\prime \prime}(t)=8 ;\) then use Newton's second law \(\left(F=m a=m x^{\prime \prime}(t)\right)\) to evaluate the work integral \(W=\int_{x_{0}}^{x_{f}} F(x) d x,\) where \(x_{0}\) and \(x_{f}\) are the initial and final positions, respectively. b. Change variables in the work integral and integrate with respect to \(t .\) Be sure your answer agrees with part (a).

Determine the following indefinite integrals. \(\int \frac{d x}{\sqrt{x^{2}-16}}\)

Kelly started at noon \((t=0)\) riding a bike from Niwot to Berthoud, a distance of \(20 \mathrm{km},\) with velocity \(v(t)=15 /(t+1)^{2}\) (decreasing because of fatigue). Sandy started at noon \((t=0)\) riding a bike in the opposite direction from Berthoud to Niwot with velocity \(u(t)=20 /(t+1)^{2}\) (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours. a. Make a graph of Kelly's distance from Niwot as a function of time. b. Make a graph of Sandy's distance from Berthoud as a function of time. c. How far has each person traveled when they meet? When do they meet? d. More generally, if the riders' speeds are \(v(t)=A /(t+1)^{2}\) and \(u(t)=B /(t+1)^{2}\) and the distance between the towns is \(D,\) what conditions on \(A, B,\) and \(D\) must be met to ensure that the riders will pass each other? e. Looking ahead: With the velocity functions given in part (d), make a conjecture about the maximum distance each person can ride (given unlimited time).

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

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{e}+e^{x}\right)$$

Find the volume of the following solids of revolution. Sketch the region in question. The region bounded by \(y^{2}=\ln x, y^{2}=\ln x^{3},\) and \(y=2\) revolved about the \(x\) -axis

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and that the chain has a density of \(5 \mathrm{kg} / \mathrm{m}\). a. How much work is required to wind the entire chain onto the cylinder using the winch? b. How much work is required to wind the chain onto the cylinder if a \(50-\mathrm{kg}\) block is attached to the end of the chain?

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(1+\frac{4}{x}\right)^{x}$$

Determine the following indefinite integrals. \(\int \frac{e^{x}}{36-e^{2 x}} d x, x<\ln 6\)

Theo and Sasha start at the same place on a straight road riding bikes with the following velocities (measured in \(\mathrm{mi} / \mathrm{hr}\) ): Theo: \(v_{T}(t)=10,\) for \(t \geq 0\), Sasha: \(v_{S}(t)=15 t,\) for \(0 \leq t \leq 1\) and \(v_{S}(t)=15,\) for \(t>1\). a. Graph the velocity functions for both riders. b. If the riders ride for \(1 \mathrm{hr},\) who rides farther? Interpret your answer geometrically using the graphs of part (a). c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). d. Which rider arrives first at the \(10-, 15\) -, and 20 -mile markers of the race? Interpret your answer geometrically using the graphs of part (a). e. Suppose Sasha gives Theo a head start of \(0.2 \mathrm{mi}\) and the riders ride for 20 mi. Who wins the race? f. Suppose Sasha gives Theo a head start of \(0.2 \mathrm{hr}\) and the riders ride for 20 mi. Who wins the race?

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. \(\int_{0}^{2} \pi\left(x^{2}+2 x+1\right) d x\)

Find the volume of the following solids using the method of your choice. The solid formed when the region bounded by \(y=x^{2}\) and \(y=2-x^{2}\) is revolved about the \(x\) -axis

A 60-m-long, 9.4-mm-diameter rope hangs free from a ledge. The density of the rope is \(55 \mathrm{g} / \mathrm{m}\). How much work is needed to pull the entire rope to the ledge?

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right)$$

Determine the following indefinite integrals. \(\int \frac{d x}{x \sqrt{16+x^{2}}}\)

At noon \((t=0),\) Alicia starts running along a long straight road at \(4 \mathrm{mi} / \mathrm{hr}\). Her velocity decreases according to the function \(v(t)=4 /(t+1),\) for \(t \geq 0 .\) At noon, Boris also starts running along the same road with a 2 -mi head start on Alicia; his velocity is given by \(u(t)=2 /(t+1),\) for \(t \geq 0\). a. Find the position functions for Alicia and Boris, where \(s=0\) corresponds to Alicia's starting point. b. When, if ever, does Alicia overtake Boris?

Find the volume of the following solids using the method of your choice. The solid formed when the region bounded by \(y=\sin x\) and \(y=1-\sin x\) between \(x=\pi / 6\) and \(x=5 \pi / 6\) is revolved about the \(x\) -axis

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) The region bounded by \(f(x)=x\) and \(g(x)=x^{n}\) for \(x \geq 0\)

A body of mass \(m\) is suspended by a rod of length \(L\) that pivots without friction (see figure). The mass is slowly lifted along a circular arc to a height \(h\) a. Assuming that the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is \(F=m g \sin \theta\) b. Noting that an element of length along the path of the pendulum is \(d s=L d \theta,\) evaluate an integral in \(\theta\) to show that the work done in lifting the mass to a height \(h\) is \(m g h\)

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(\cos \left(x^{2 \sin x}\right)\right)$$

Determine the following indefinite integrals. \(\int \frac{d x}{x \sqrt{4-x^{8}}}\)

A strong west wind blows across a circular running track. Abe and Bess start at the south end of the track and at the same time, Abe starts running clockwise and Bess starts running counterclockwise. Abe runs with a speed (in units of miles/hour) given by \(u(\varphi)=3-2 \cos \varphi\) and Bess runs with a speed given by \(v(\theta)=3+2 \cos \theta,\) where \(\varphi\) and \(\theta\) are the central angles of the runners. a. Graph the speed functions \(u\) and \(v,\) and explain why they describe the runners' speeds (in light of the wind). b. Compute each runner's average speed (over one lap) with respect to the central angle. c. Challenge: If the track has a radius of \(\frac{1}{10} \mathrm{mi}\), how long does it take each runner to complete one lap and who wins the race?

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 \(V_{C}\)

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) The region bounded by \(f(x)=x\) and \(g(x)=x^{1 / n}\) for \(x \geq 0\)

Evaluate the following integrals. $$\int 7^{2 x} d x$$

A \(200-\mathrm{L}\) cistern is empty when water begins flowing into it (at \(t=0\) ) at a rate (in liters/minute) given by \(Q^{\prime}(t)=3 \sqrt{t}\). a. How much water flows into the cistern in 1 hour? b. Find and graph the function that gives the amount of water in the tank at any time \(t \geq 0\). c. When will the tank be full?

Find the volume of the following solids using the method of your choice. The solid formed when the region bounded by \(y=x^{3}\), the \(x\) -axis, and \(x=2\) is revolved about the \(x\) -axis

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 \(h=0 \text { and } h=8 .)\)

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) The region bounded by \(f(x)=x^{1 / n}\) and \(g(x)=x^{n}\) for \(x \geq 0\)

Determine the following indefinite integrals. \(\int \frac{d x}{x \sqrt{1+x^{4}}}\)

Evaluate the following integrals. $$\int 3^{-2 x} d x$$

Suppose that \(r(t)=r_{0} e^{-k t},\) with \(k>0,\) is the rate at which a nation extracts oil, where \(r_{0}=10^{7}\) barrels \(/\) yr is the current rate of extraction. Suppose also that the estimate of the total oil reserve is \(2 \times 10^{9}\) barrels. a. Find \(Q(t),\) the total amount of oil extracted by the nation after \(t\) years. b. Evaluate \(\lim _{t \rightarrow \infty} Q(t)\) and explain the meaning of this limit. c. Find the minimum decay constant \(k\) for which the total oil reserves will last forever. d. Suppose \(r_{0}=2 \times 10^{7}\) barrels/yr and the decay constant \(k\) is the minimum value found in part (c). How long will the total oil reserves last?

Find the volume of the following solids using the method of your choice. The solid whose base is the region bounded by \(y=x^{2}\) and the line \(y=1\) and whose cross sections perpendicular to the base and parallel to the \(x\) -axis are semicircles

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.

Find the area of the following regions, expressing your results in terms of the positive integer \(n \geq 2\) Let \(A_{n}\) be the area of the region bounded by \(f(x)=x^{1 / n}\) and \(g(x)=x^{n}\) on the interval \([0,1],\) where \(n\) is a positive integer. Evaluate \(\lim _{n \rightarrow \infty} A_{n}\) and interpret the result.

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 \(1 \mathrm{g} / \mathrm{cm}^{3}(1 \mathrm{Cal}=4184 \mathrm{J})\)

Evaluate the following integrals. $$\int_{0}^{5} 5^{5 x} d x$$

Let \(R\) be the region bounded by \(y=x^{2}\) and \(y=\sqrt{x} .\) Which is greater, the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or about the line \(y=1 ?\)

Find the volume of the following solids using the method of your choice. The solid formed when the region bounded by \(y=2, y=2 x+2\), and \(x=6\) is revolved about the \(y\) -axis

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?

Evaluate the following definite integrals. Use Theorem 10 to express your answer in terms of logarithms. \(\int_{-2}^{2} \frac{d t}{t^{2}-9}\)

A Lorenz curve is given by \(y=L(x),\) where \(0 \leq x \leq 1\) represents the lowest fraction of the population of a society in terms of wealth and \(0 \leq y \leq 1\) represents the fraction of the total wealth that is owned by that fraction of the society. For example, the Lorenz curve in the figure shows that \(L(0.5)=0.2,\) which means that the lowest \(0.5(50 \%)\) of the society owns \(0.2(20 \%)\) of the wealth. a. A Lorenz curve \(y=L(x)\) is accompanied by the line \(y=x\) called the line of perfect equality. Explain why this line is given this name. b. Explain why a Lorenz curve satisfies the conditions \(L(0)=0, L(1)=1,\) and \(L^{\prime}(x) \geq 0\) on [0,1] c. Graph the Lorenz curves \(L(x)=x^{p}\) corresponding to \(p=1.1,1.5,2,3,4 .\) Which value of \(p\) corresponds to the most equitable distribution of wealth (closest to the line of perfect equality)? Which value of \(p\) corresponds to the least equitable distribution of wealth? Explain. d. The information in the Lorenz curve is often summarized in a single measure called the Gini index, which is defined as follows. Let \(A\) be the area of the region between \(y=x\) and \(y=L(x)\) (see figure) and let \(B\) be the area of the region between \(y=L(x)\) and the \(x\) -axis. Then the Gini index is $$G=\frac{A}{A+B} . \text { Show that } G=2 A=1-2 \int_{0}^{1} L(x) d x$$ e. Compute the Gini index for the cases \(L(x)=x^{p}\) and \(p=1.1,1.5,2,3,4\) f. What is the smallest interval \([a, b]\) on which values of the Gini index lie for \(L(x)=x^{p}\) with \(p \geq 1 ?\) Which endpoints of \([a, b]\) correspond to the least and most equitable distribution of wealth? g. Consider the Lorenz curve described by \(L(x)=5 x^{2} / 6+x / 6\) Show that it satisfies the conditions \(L(0)=0, L(1)=1,\) and \(L^{\prime}(x) \geq 0\) on \([0,1] .\) Find the Gini index for this function.

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 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).

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

Evaluate the following definite integrals. Use Theorem 10 to express your answer in terms of logarithms. \(\int_{1 / 6}^{1 / 4} \frac{d t}{t \sqrt{1-4 t^{2}}}\)

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

Consider the region \(R\) in the first quadrant bounded by \(y=x^{1 / n}\) and \(y=x^{n}\), where \(n\) 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.