Chapter 6

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. $$y=\frac{1}{x} \text { on }[1,10]$$

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

Sketch each region (if a figure is not given) and find its area by integrating with respect to \(y\) The region bounded by \(x=y^{2}-3 y+12\) and \(x=-2 y^{2}-6 y+30\)

Use the method of your choice to determine the area of the surface generated when the following curves are revolved about the indicated axis. \(y=9 x^{2 / 3}-\frac{x^{4 / 3}}{32},\) for \(1 \leq x \leq 8 ;\) about the \(x\) -axis

A large die-casting machine used to make automobile engine blocks is purchased for \(\$ 2.5\) million. For tax purposes, the value of the machine can be depreciated by \(6.8 \%\) of its current value each year. a. What is the value of the machine after 10 years? b. After how many years is the value of the machine \(10 \%\) of its original value?

A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in \(\mathrm{m} / \mathrm{s}\) ) given by \(v(t)=9.8 t\) neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of \(10 \mathrm{m} / \mathrm{s},\) which it maintains until it enters the ocean. a. Graph the velocity function. b. How far does the probe fall in the first 30 s after it is released? c. If the probe was released from an altitude of \(3 \mathrm{km},\) when does it enter the ocean?

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. \(y=\sqrt{50-2 x^{2}},\) in the first quadrant

A spring has a restoring force given by \(F(x)=25 x .\) Let \(W(x)\) be the work required to stretch the spring from its equilibrium position \((x=0)\) to a variable distance \(x\). Find and graph the work function. Compare the work required to stretch the spring \(x\) units from equilibrium to the work required to compress the spring \(x\) units from equilibrium.

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

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. $$y=\frac{1}{x^{2}+1} \text { on }[-5,5]$$

Sketch each region (if a figure is not given) and find its area by integrating with respect to \(y\) Both regions bounded by \(x=y^{3}-4 y^{2}+3 y\) and \(x=y^{2}-y\)

Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=x^{5} \text { on }[0,1]$$

The pressure of Earth's atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of 30,000 ft (approximately the altitude of Mt. Everest), the pressure is one- third the sea-level pressure. At what elevation is the pressure half the sea- level pressure? At what elevation is it \(1 \%\) of the sea-level pressure?

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-32, v(0)=70, s(0)=10$$

Integrals with general bases Evaluate the following integrals. \(\int_{-1}^{1} 10^{x} d x\)

Find the arc length of the following curves by integrating with respect to \(y\) \(x=2 y-4,\) for \(-3 \leq y \leq 4\) (Use calculus.)

Compute \(dy/dx\) for the following functions. $$y=\ln \operatorname{sech} 2 x$$

Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=\cos x \text { on }\left[0, \frac{\pi}{2}\right]$$

Use the shell method to find the volume of the following solids. A right circular cone of radius 3 and height 8

The half-life of \(\mathrm{C}-14\) is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only \(77 \%\) of the C-14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has \(6.2 \%\) of the \(\mathrm{C}-14\) that it had when it was alive. Estimate when the wood was cut.

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-32, v(0)=50, s(0)=0$$

A cylindrical water tank has height 8 m and radius 2 m (see figure). a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank? b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.

Integrals with general bases Evaluate the following integrals. \(\int_{0}^{\pi / 2} 4^{\sin x} \cos x d x\)

Find the arc length of the following curves by integrating with respect to \(y\) $$y=\ln (x-\sqrt{x^{2}-1}), \text { for } 1 \leq x \leq \sqrt{2}$$

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

Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=\ln x^{2} \text { on }[1, \sqrt{e}]$$

Use the shell method to find the volume of the following solids. The solid formed when a hole of radius 2 is drilled symmetrically along the axis of a right circular cylinder of height 6 and radius 4.

Uranium- 238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and determine that \(85 \%\) of the original \(U-238\) remains; the other \(15 \%\) has decayed into lead. How old is the rock?

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-9.8, v(0)=20, s(0)=0$$

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=e^{x / 2}, y=e^{-x / 2}, x=\ln 2, x=\ln 3$$

Integrals with general bases Evaluate the following integrals. \(\int_{1}^{2}(1+\ln x) x^{x} d x\)

Compute \(dy/dx\) for the following functions. $$y=x^{2} \cosh ^{2} 3 x$$

Consider the following curves on the given intervals. a. Write the integral that gives the area of the surface generated when the curve is revolved about the \(x\) -axis. b. Use a calculator or software to approximate the surface area. $$y=\tan x \text { on }\left[0, \frac{\pi}{4}\right]$$

Use the shell method to find the volume of the following solids. The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9.

Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for \(1 \%\) of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or I-131. This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries. a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of \(I-131\) in the body after \(t \geq 0\) days. b. How long does it take the amount of I-131 to reach 10\% of the initial dose? c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach \(10 \%\) of the initial dose change if the initial dose is increased by \(5 \% ?\)

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=e^{-t}, v(0)=60, s(0)=40$$

Let \(R\) be the region bounded by the following curves. Use the washer method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. $$y=x, y=x+2, x=0, x=4$$

Integrals with general bases Evaluate the following integrals. \(\int_{1 / 3}^{1 / 2} \frac{10^{1 / p}}{p^{2}} d p\)

Find the arc length of the following curves by integrating with respect to \(y\) $$x=2 e^{\sqrt{2} y}+\frac{1}{16} e^{-\sqrt{2} y}, \text { for } 0 \leq y \leq \frac{\ln 2}{\sqrt{2}}$$

The volume of a cone of radius \(r\) and height \(h\) is one-third the volume of a cylinder with the same radius and height. Does the surface area of a cone of radius \(r\) and height \(h\) equal one-third the surface area of a cylinder with the same radius and height? If not, find the correct relationship. Exclude the bases of the cone and cylinder.

Use the shell method to find the volume of the following solids. The solid formed when a hole of radius 3 is drilled symmetrically through the center of a sphere of radius 6

Determine whether the following statements are true and give an explanation or counterexample. a. A quantity that increases at \(6 \% /\) yr obeys the growth function $$y(t)=y_{0} e^{0.06 t}.$$ b. If a quantity increases by \(10 \% / \mathrm{yr},\) it increases by \(30 \%\) over 3 years. c. A quantity decreases by one-third every month. Therefore, it decreases exponentially. d. If the rate constant of an exponential growth function is increased, its doubling time is decreased. e. If a quantity increases exponentially, the time required to increase by a factor of 10 remains constant for all time.

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=-0.01 t, v(0)=10, s(0)=0$$

Determine each indefinite integral. $$\int \cosh 2 x d x$$

Consider the upper half of the astroid described by \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3},\) where \(a>0\) and \(|x| \leq a\). Find the area of the surface generated when this curve is revolved about the \(x\) -axis. Use symmetry. Note that the function describing the curve is not differentiable at \(0 .\) However, the surface area integral can be evaluated using methods you know.

Use the shell method to find the volume of the following solids. The ellipsoid formed when that part of the ellipse \(x^{2}+2 y^{2}=4\) with \(x \geq 0\) is revolved about the \(y\) -axis

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=\frac{20}{(t+2)^{2}}, v(0)=20, s(0)=10$$

A swimming pool is \(20 \mathrm{m}\) long and \(10 \mathrm{m}\) wide, with a bottom that slopes uniformly from a depth of \(1 \mathrm{m}\) at one end to a depth of \(2 \mathrm{m}\) at the other end (see figure). Assuming the pool is full, how much work is required to pump the water to a level \(0.2 \mathrm{m}\) above the top of the pool?

Integrals with general bases Evaluate the following integrals. \(\int \frac{4^{\cot x}}{\sin ^{2} x} d x\)

Consider the segment of the line \(y=m x+c\) on the interval \([a, b] .\) Use the arc length formula to show that the length of the line segment is \((b-a) \sqrt{1+m^{2}}\) Verify this result by computing the length of the line segment using the distance formula.