Chapter 6

Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=x^{-1}, y=4 x\), and \(y=x / 4\)

Find the derivatives of the following functions. $$f(x)=\operatorname{csch}^{-1}(2 / x)$$

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

Hooke's law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by \(F(x)=16 x-0.1 x^{3},\) for \(|x| \leq 7\) a. Graph the restoring force and interpret it. b. How much work is done in stretching the spring from its equilibrium position \((x=0)\) to \(x=1.5 ?\) c. How much work is done in compressing the spring from its equilibrium position \((x=0)\) to \(x=-2 ?\)

Let \(f(x)=x^{p}\) and \(g(x)=x^{1 / q},\) where \(p>1\) and \(q>1\) are positive integers. Let \(R_{1}\) be the region in the first quadrant between \(y=f(x)\) and \(y=x\) and let \(R_{2}\) be the region in the first quadrant between \(y=g(x)\) and \(y=x\) a. Find the area of \(R_{1}\) and \(R_{2}\) when \(p=q,\) and determine which region has the greater area. b. Find the area of \(R_{1}\) and \(R_{2}\) when \(p>q,\) and determine which region has the greater area. c. Find the area of \(R_{1}\) and \(R_{2}\) when \(p

Find the derivatives of the following functions. $$f(x)=x \sinh ^{-1} x-\sqrt{x^{2}+1}$$

Compute the following derivatives using the method of your choice. \(\frac{d}{d x}\left(e^{-10 x^{2}}\right)\)

Find the volume of the solid generated in the following situations. The region \(R\) is bounded by the graph of \(f(x)=2 x(2-x)\) and the \(x\) -axis. Which is greater, the volume of the solid generated when \(R\) is revolved about the line \(y=2\) or the volume of the solid generated when \(R\) is revolved about the line \(y=0 ?\) Use integration to justify your answer.

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 _____. $$v(t)=2 t+6, \text { for } 0 \leq t \leq 8$$

\(\mathrm{A} 10-\mathrm{kg}\) mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with \(F(x)=k x\) a. How much work is required to compress the spring and lift the mass 0.5 m? b. How much work is required to stretch the spring and lower the mass 0.5 m?

Find the derivatives of the following functions. $$f(u)=\sinh ^{-1}(\tan u)$$

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

Find the volume of the following solids of revolution. Sketch the region in question. The region bounded by \(y=1 /\left(x^{2}+1\right), y=0, x=1,\) and \(x=4\) revolved about the \(y\) -axis

Determine whether the following statements are true and give an explanation or counterexample. a. A pyramid is a solid of revolution. b. The volume of a hemisphere can be computed using the disk method. c. Let \(R_{1}\) be the region bounded by \(y=\cos x\) and the \(x\) -axis on \([-\pi / 2, \pi / 2] .\) Let \(R_{2}\) be the region bounded by \(y=\sin x\) and the \(x\) -axis on \([0, \pi] .\) The volumes of the solids generated when \(R_{1}\) and \(R_{2}\) are revolved about the \(x\) -axis are equal.

A glass has circular cross sections that taper (linearly) from a radius of \(5 \mathrm{cm}\) at the top of the glass to a radius of \(4 \mathrm{cm}\) at the bottom. The glass is \(15 \mathrm{cm}\) high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is \(5 \mathrm{cm}\) above the top of the glass? Assume the density of orange juice equals the density of water.

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

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=(\ln x) / \sqrt{x}, y=0,\) and \(x=2\) revolved about the \(x\) -axis

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 _____. $$v(t)=2 \sin t, \text { for } 0 \leq t \leq \pi$$

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

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=1 / \sqrt{x}, y=0, x=2,\) and \(x=6\) revolved about the \(x\) -axis

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

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

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=\frac{1}{\sqrt{x^{2}+1}}\) and \(y=\frac{1}{\sqrt{2}}\) revolved about the \(x\) -axis

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. When do they meet? How far has each person traveled when 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 \(\vec{D},\) 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).

A rigid body with a mass of 2 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 \(5 \mathrm{s}\) 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 \(W=\int_{x_{0}}^{x_{1}} F(x) d x,\) 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).

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. $$\int \frac{d x}{x \sqrt{16+x^{2}}}$$

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 volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{x}, y=0, x=0,\) and \(x=2\) revolved about the \(x\) -axis

Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in \(\mathrm{mi} / \mathrm{hr}\) ). Assume \(t\) is measured in hours. 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 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?

A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and 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. \(\frac{d}{d x}\left(\cos \left(x^{2 \sin x}\right)\right)\)

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

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{-x}, y=e^{x}, x=0,\) and \(x=\ln 4\) revolved about the \(x\) -axis

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 \(t \geq 0 .\) Assume \(t\) is measured in hours. 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 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 \(x \geq 0\)

Miscellaneous integrals Evaluate the following integrals. \(\int 7^{2 x} d x\)

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

A strong west wind blows across a circular running track. Abe and Bess start running 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 \(\mathrm{mi} / \mathrm{hr}\) ) given by \(u(\varphi)=3-2 \cos \varphi\) and \(\mathrm{Bess}\) 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 the average value of \(u\) and \(v\) 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 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 \(m g h\)

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.

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

Miscellaneous integrals Evaluate the following integrals. \(\int 3^{-2 x} d x\)

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 solid of revolution. Sketch the region in question. The region bounded by \(y=e^{-x}, y=0, x=0,\) and \(x=p>0\) revolved about the \(x\) -axis (Is the volume bounded as \(p \rightarrow \infty ?\))

\(A\) 2000-liter cistern is empty when water begins flowing into it (at \(t=0\) ) at a rate (in \(\mathrm{L} / \mathrm{min}\) ) given by \(Q^{\prime}(t)=3 \sqrt{t},\) where \(t\) is measured in minutes. 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?

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 \(y=1-x,\) the \(x\) -axis, and the \(y\) -axis.

Definite integrals Evaluate the following definite integrals. Use Theorem \(6.10\) to express your answer in terms of logarithms. $$\int_{5}^{3 \sqrt{5}} \frac{d x}{\sqrt{x^{2}-9}}$$

Miscellaneous integrals Evaluate the following integrals. \(\int_{0}^{5} 5^{5 x} d x\)

Let \(R\) be the region bounded by the curve \(y=\sqrt{x+a}(\text { with } a>0),\) 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 \((T)=\frac{8}{5}\).