Chapter 6

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=x, y=2-x, \text { and } y=0$$

Evaluate the following integrals. Include absolute values only when needed. \(\int_{e^{2}}^{e^{3}} \frac{d x}{x \ln x \ln ^{2}(\ln x)}\)

Suppose a force of \(30 \mathrm{N}\) is required to stretch and hold a spring \(0.2 \mathrm{m}\) from its equilibrium position. a. Assuming the spring obeys Hooke's law, find the spring constant \(k\) b. How much work is required to compress the spring \(0.4 \mathrm{m}\) from its equilibrium position? c. How much work is required to stretch the spring \(0.3 \mathrm{m}\) from its equilibrium position? d. How much additional work is required to stretch the spring \(0.2 \mathrm{m}\) if it has already been stretched \(0.2 \mathrm{m}\) from its equilibrium position?

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\ln x \text { on }[1,4]$$

Derive the following derivative formulas given that \(d / d x(\cosh x)=\sinh x\) and \(d / d x(\sinh x)=\cosh x.\) $$d / d x(\operatorname{coth} x)=-\operatorname{csch}^{2} x$$

Sketch each region (if a figure is not given) and then find its total area. The region bounded by \(y=2-|x|\) and \(y=x^{2}\)

Starting in \(2010(t=0),\) the rate at which oil is consumed by a small country increases at a rate of \(1.5 \% / \mathrm{yr},\) starting with an initial rate of 1.2 million barrels/yr. a. How much oil is consumed over the course of the year 2010 (between \(t=0\) and \(t=1\) )? b. Find the function that gives the amount of oil consumed between \(t=0\) and any future time \(t\). c. How many years after 2010 will the amount of oil consumed since 2010 reach 10 million barrels?

Let \(R\) be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. \(y=\cos x\) on \([0, \pi / 2], y=0, x=0\) (Recall that \(\cos ^{2} x= \frac{1}{2}(1+\cos 2 x).)\)

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=1 /(t+1) \text { on }[0,8] ; s(0)=-4$$

Suppose a force of \(15 \mathrm{N}\) is required to stretch and hold a spring \(0.25 \mathrm{m}\) from its equilibrium position. a. Assuming the spring obeys Hooke's law, find the spring constant \(k\) b. How much work is required to compress the spring \(0.2 \mathrm{m}\) from its equilibrium position? c. How much additional work is required to stretch the spring \(0.3 \mathrm{m}\) if it has already been stretched \(0.25 \mathrm{m}\) from its equilibrium position?

Evaluate the following integrals. Include absolute values only when needed. \(\int_{0}^{1} \frac{y \ln ^{4}\left(y^{2}+1\right)}{y^{2}+1} d y\)

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\frac{x^{3}}{3} \text { on }[-1,1]$$

Derive the following derivative formulas given that \(d / d x(\cosh x)=\sinh x\) and \(d / d x(\sinh x)=\cosh x.\) $$d / d x(\operatorname{sech} x)=-\operatorname{sech} x \tanh x$$

Find the area of the surface generated when the given curve is revolved about the \(y\) -axis. The part of the curve \(y=\frac{1}{2} \ln (2 x+\sqrt{4 x^{2}-1})\) between the points \(\left(\frac{1}{2}, 0\right)\) and \(\left(\frac{17}{16}, \ln 2\right)^{4}\).

Sketch each region (if a figure is not given) and then find its total area. The regions bounded by \(y=x^{3}\) and \(y=9 x\)

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. The homicide rate decreases at a rate of \(3 \% / \mathrm{yr}\) in a city that had 800 homicides/yr in \(2010 .\) At this rate, when will the homicide rate reach 600 homicides/yr?

Let \(R\) be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. \(y=\sin x \text { on }[0, \pi], y=0\) (Recall that \(\sin ^{2} x=\frac{1}{2}(1-\cos 2 x) .)\)

A mass hanging from a spring is set in motion, and its ensuing velocity is given by \(v(t)=2 \pi \cos \pi t\) for \(t \geq 0 .\) Assume that the positive direction is upward and that \(s(0)=0\) a. Determine the position function, for \(t \geq 0\) b. Graph the position function on the interval [0,4] c. At what times does the mass reach its low point the first three times? d. At what times does the mass reach its high point the first three times?

A spring on a horizontal surface can be stretched and held \(0.5 \mathrm{m}\) from its equilibrium position with a force of \(50 \mathrm{N}\) a. How much work is done in stretching the spring \(1.5 \mathrm{m}\) from its equilibrium position? b. How much work is done in compressing the spring \(0.5 \mathrm{m}\) from its equilibrium position?

Evaluate the following integrals. \(\int_{0}^{2} 4 x e^{-x^{2} / 2} d x\)

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\sqrt{x-2} \text { on }[3,4]$$

Derive the following derivative formulas given that \(d / d x(\cosh x)=\sinh x\) and \(d / d x(\sinh x)=\cosh x.\) $$d / d x(\operatorname{csch} x)=-\operatorname{csch} x \operatorname{coth} x$$

Determine whether the following statements are true and give an explanation or counterexample. a. If the curve \(y=f(x)\) on the interval \([a, b]\) is revolved about the \(y\) -axis, the area of the surface generated is $$\int_{f(a)}^{f(b)} 2 \pi f(y) \sqrt{1+f^{\prime}(y)^{2}} d y$$ b. If \(f\) is not one-to-one on the interval \([a, b],\) then the area of the surface generated when the graph of \(f\) on \([a, b]\) is revolved about the \(x\) -axis is not defined. c. Let \(f(x)=12 x^{2} .\) The area of the surface generated when the graph of \(f\) on [-4,4] is revolved about the \(x\) -axis is twice the area of the surface generated when the graph of \(f\) on [0,4] is revolved about the \(x\) -axis. d. Let \(f(x)=12 x^{2} .\) The area of the surface generated when the graph of \(f\) on [-4,4] is revolved about the \(y\) -axis is twice the area of the surface generated when the graph of \(f\) on [0,4] is revolved about the \(y\) -axis.

Sketch each region (if a figure is not given) and then find its total area. The region bounded by \(y=|x-3|\) and \(y=x / 2\)

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. A drug is eliminated from the body at a rate of \(15 \% / \mathrm{hr}\). After how many hours does the amount of drug reach \(10 \%\) of the initial dose?

A cyclist rides down a long straight road at a velocity (in \(\mathrm{m} / \mathrm{min}\) ) given by \(v(t)=400-20 t,\) for \(0 \leq t \leq 10\) where \(t\) is measured in minutes. a. How far does the cyclist travel in the first 5 min? b. How far does the cyclist travel in the first 10 min? c. How far has the cyclist traveled when her velocity is \(250 \mathrm{m} / \mathrm{min} ?\)

Evaluate the following integrals. \(\int \frac{e^{\sin x}}{\sec x} d x\)

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\frac{8}{x^{2}} \text { on }[1,4]$$

Compute \(dy/dx\) for the following functions. $$y=\sinh 4 x$$

Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. \(x=\sqrt{12 y-y^{2}},\) for \(2 \leq y \leq 10 ;\) about the \(y\) -axis

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=x^{3}, y=1, \text { and } x=0$$

Sketch each region (if a figure is not given) and then find its total area. The regions bounded by \(y=x^{2}(3-x)\) and \(y=12-4 x\)

The velocity (in \(\mathrm{mi} / \mathrm{hr}\) ) of an airplane flying into a headwind is given by \(v(t)=30\left(16-t^{2}\right),\) for \(0 \leq t \leq 3 .\) Assume that \(s(0)=0\) and \(t\) is measured in hours. a. Determine and graph the position function, for \(0 \leq t \leq 3\) b. How far does the airplane travel in the first \(2 \mathrm{hr} ?\) c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

Calculate the work required to stretch the following springs \(0.5 \mathrm{m}\) from their equilibrium positions. Assume Hooke's law is obeyed. a. \(\mathrm{A}\) spring that requires a force of \(50 \mathrm{N}\) to be stretched \(0.2 \mathrm{m}\) from its equilibrium position b. A spring that requires \(50 \mathrm{J}\) of work to be stretched \(0.2 \mathrm{m}\) from its equilibrium position

Evaluate the following integrals. \(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x\)

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=\cos 2 x \text { on }[0, \pi]$$

Compute \(dy/dx\) for the following functions. $$y=\cosh ^{2} x$$

Sketch each region (if a figure is not given) and find its area by integrating with respect to \(y\) The region bounded by \(y=\sqrt{\frac{x}{2}+1}, y=\sqrt{1-x}\), and \(y=0\)

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. China's one-child policy was implemented with a goal of reducing China's population to 700 million by 2050 (from 1.2 billion in 2000 ). Suppose China's population declines at a rate of \(0.5 \% /\) yr. Will this rate of decline be sufficient to meet the goal?

The velocity (in \(\mathrm{mi} / \mathrm{hr}\) ) of a hiker walking along a straight trail is given by \(v(t)=3 \sin ^{2}(\pi t / 2),\) for \(0 \leq t \leq 4\) Assume that \(s(0)=0\) and \(t\) is measured in hours. a. Determine and graph the position function, for \(0 \leq t \leq 4\) (Hint: \(\sin ^{2} t=\frac{1}{2}(1-\cos 2 t)\) b. What is the distance traveled by the hiker in the first 15 min of the hike? c. What is the hiker's position at \(t=3 ?\)

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=\sqrt{\sin ^{-1} x}, y=\sqrt{\pi / 2}, \text { and } x=0$$

Calculate the work required to stretch the following springs \(0.4 \mathrm{m}\) from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires a force of \(50 \mathrm{N}\) to be stretched \(0.1 \mathrm{m}\) from its equilibrium position b. A spring that requires 2 J of work to be stretched \(0.1 \mathrm{m}\) from its equilibrium position

Evaluate the following integrals. \(\int_{-2}^{2} \frac{e^{z / 2}}{e^\)\int_{-2}^{2} \frac{e^{z / 2}}{1+e^{z / 2}} d z=2\({z / 2}+1} d z\)

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=4 x-x^{2} \text { on }[0,4]$$

Compute \(dy/dx\) for the following functions. $$y=-\sinh ^{3} 4 x$$

Sketch each region (if a figure is not given) and find its area by integrating with respect to \(y\) The region bounded by \(x=\cos y\) and \(x=-\sin 2 y\)

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. The population of Michigan decreased from 9.94 million in 2000 to 9.88 million in \(2010 .\) Use an exponential model to predict the population in 2020 . Explain why an exponential (decay) model might not be an appropriate long-term model of the population of Michigan.

The velocity of a (fast) automobile on a straight highway is given by the function $$v(t)=\left\\{\begin{array}{ll} 3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45 \end{array}\right.$$ where \(t\) is measured in seconds and \(v\) has units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function, for \(0 \leq t \leq 70 .\) When is the velocity a maximum? When is the velocity zero? b. What is the distance traveled by the automobile in the first \(30 \mathrm{s} ?\) c. What is the distance traveled by the automobile in the first 60 s? d. What is the position of the automobile when \(t=75 ?\)

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. \(y=\sqrt{\cos ^{-1} x}\), in the first quadrant

Evaluate the following integrals. \(\int \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} d x\)