Chapter 6

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. 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 for 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 \(5 \% ?\)

Compute \(d y / d x\) for the following functions. \(y=x / \operatorname{csch} x\)

Function defined as an integral Write the integral that gives the length of the curve \(y=f(x)=\int_{0}^{x} \sin t d t\) on the interval \([0, \pi].\)

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.

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. $$y=x, y=x+2, x=0, x=4$$

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

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

Explain why or why not 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.

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. $$a(t)=\cos 2 t, v(0)=5, s(0)=7$$

Find the area of the following regions by (a) integrating with respect to \(x,\) and (b) integrating with respect to \(y\) Be sure your results agree. Sketch the bounding curves and the region in question. The region bounded by \(y=2 x-1\) and \(x=y^{2}\)

A water tank is shaped like an inverted cone with height \(6 \mathrm{m}\) and base radius \(1.5 \mathrm{m}\) (see figure). a. If the tank is full, 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 filled to half its depth as when it is full? Explain.

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?

Evaluate the following integrals. $$\int \frac{4^{\cot x}}{\sin ^{2} x} d x$$

Determine each indefinite integral. \(\int \operatorname{sech}^{2} x \tanh x d x\)

When the circle \(x^{2}+(y-a)^{2}=r^{2}\) on the interval \([-r, r]\) is revolved about the \(x\) -axis, the result is the surface of a torus, where \(0

Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. $$a(t)=\frac{2 t}{\left(t^{2}+1\right)^{2}}, v(0)=0, s(0)=0$$

A quantity increases according to the exponential function \(y(t)=y_{0} e^{k t} .\) What is the tripling time for the quantity? What is the time required for the quantity to increase \(p\) -fold?

Use the shell method to find the volume of the following solids. A hole of radius \(r \leq R\) is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola \(y=6\left(1-\frac{x^{2}}{R^{2}}\right)\) about the \(y\) -axis, where \(0 \leq x \leq R\).

Evaluate the derivatives of the following functions. $$f(x)=(2 x)^{4 x}$$

Prove that the doubling time for an exponentially increasing quantity is constant for all time.

Determine each indefinite integral. \(\int \frac{\sinh x}{1+\cosh x} d x\)

Suppose the graph of \(f\) on the interval \([a, b]\) has length \(L,\) where \(f^{\prime}\) is continuous on \([a, b] .\) Evaluate the following integrals in terms of \(L\) a. \(\int_{a / 2}^{b / 2} \sqrt{1+f^{\prime}(2 x)^{2}} d x \quad\) b. \(\int_{a / c}^{b / c} \sqrt{1+f^{\prime}(c x)^{2}} d x\) if \(c \neq 0.\)

Suppose a sphere of radius \(r\) is sliced by two horizontal planes \(h\) units apart (see figure). Show that the surface area of the resulting zone on the sphere is \(2 \pi r h,\) independent of the location of the cutting planes.

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. $$y=\sin x, y=\sqrt{\sin x}, \text { for } 0 \leq x \leq \pi / 2$$

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

A drag racer accelerates at \(a(t)=88 \mathrm{ft} / \mathrm{s}^{2} .\) Assume that \(v(0)=0\) and \(s(0)=0\). a. Determine and graph the position function, for \(t \geq 0\). b. How far does the racer travel in the first 4 seconds? c. At this rate, how long will it take the racer to travel \(\frac{1}{4}\) mi? d. How long does it take the racer to travel \(300 \mathrm{ft} ?\) e. How far has the racer traveled when it reaches a speed of \(178 \mathrm{ft} / \mathrm{s} ?\)

Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question. The region in the first quadrant bounded by \(y=x^{2 / 3}\) and \(y=4\)

Evaluate the derivatives of the following functions. $$f(x)=x^{\pi}$$

A has a current population of 500,000 people and grows at a rate of \(3 \% /\) yr. City \(B\) has a current population of 300,000 and grows at a rate of \(5 \% / \mathrm{yr} .\) a. When will the cities have the same population? b. Suppose City C has a current population of \(y_{0}<500,000\) and a growth rate of \(p>3 \% /\) yr. What is the relationship between \(y_{0}\) and \(p\) such that the Cities \(A\) and \(C\) have the same population in 10 years?

Determine each indefinite integral. \(\int \operatorname{coth}^{2} x \operatorname{csch}^{2} x d x\)

Suppose a curve is described by \(y=f(x)\) on the interval \([-b, b],\) where \(f^{\prime}\) is continuous on \([-b, b] .\) Show that if \(f\) is symmetric about the origin ( \(f\) is odd) or \(f\) is symmetric about the \(y\) -axis ( \(f\) is even), then the length of the curve \(y=f(x)\) from \(x=-b\) to \(x=b\) is twice the length of the curve from \(x=0\) to \(x=b .\) Use a geometric argument and then prove it using integration.

If the top half of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is revolved about the \(x\) -axis, the result is an ellipsoid whose axis along the \(x\) -axis has length \(2 a,\) whose axis along the \(y\) -axis has length \(2 b,\) and whose axis perpendicular to the \(x y\) -plane has length \(2 b .\) We assume that \(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. $$y=|x|, y=2-x^{2}$$

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

A car slows down with an acceleration of \(a(t)=-15 \mathrm{ft} / \mathrm{s}^{2} .\) Assume that \(v(0)=60 \mathrm{ft} / \mathrm{s}\) and \(s(0)=0\). a. Determine and graph the position function, for \(t \geq 0\) b. How far does the car travel in the time it takes to come to rest?

Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question. The region in the first quadrant bounded by \(y=2\) and \(y=2 \sin x\) on the interval \([0, \pi / 2]\)

Evaluate the derivatives of the following functions. $$h(x)=2^{\left(x^{2}\right)}$$

Starting at the same time and place, Abe and Bob race, running at velocities \(u(t)=4 /(t+1) \mathrm{mi} / \mathrm{hr}\) and \(v(t)=4 e^{-t / 2} \mathrm{mi} / \mathrm{hr},\) respectively, for \(t \geq 0\) a. Who is ahead after \(t=5 \mathrm{hr}\) ? After \(t=10 \mathrm{hr} ?\) b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?

Determine each indefinite integral. \(\int \tanh ^{2} x d x\) (Hint: Use an identity.)

A family of exponential functions a. Show that the arc length integral for the function $$ f(x)=A e^{a x}+\frac{1}{4 A a^{2}} e^{-a x}, \text { where } a>0 \text { and } A>0 $$ may be integrated using methods you already know. b. Verify that the arc length of the curve \(y=f(x)\) on the interval \([0, \ln 2]\) is $$ A\left(2^{a}-1\right)-\frac{1}{4 a^{2} A}\left(2^{-a}-1\right) $$

In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly. a. What is the SAV ratio of a cube with side lengths \(a ?\) b. What is the SAV ratio of a ball with radius \(a ?\) c. Use the result of Exercise 34 to find the SAV ratio of an ellipsoid whose long axis has length \(2 \sqrt[3]{4 a}\), for \(a \geq 1\), and whose other two axes have half the length of the long axis. (This scaling is used so that, for a given value of \(a\), the volumes of the ellipsoid and the ball of radius \(a\) are equal.) The volume of a general ellipsoid is \(V=\frac{4 \pi}{3} A B C,\) where the axes have lengths \(2 A, 2 B,\) and \(2 C\) d. Graph the SAV ratio of the ball of radius \(a \geq 1\) as a function of \(a\) (part (b)) and graph the ellipsoid described in part (c) on the same set of axes. Which object has the smaller SAV ratio? e. Among all ellipsoids of a fixed volume, which one would you choose for the shape of an animal if the goal is to minimize heat loss?

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

At \(t=0,\) a train approaching a station begins decelerating from a speed of \(80 \mathrm{mi} / \mathrm{hr}\) according to the acceleration function \(a(t)=-1280(1+8 t)^{-3},\) where \(t \geq 0\) is measured in hours. How far does the train travel between \(t=0\) and \(t=0.2 ?\) Between \(t=0.2\) and \(t=0.4 ?\) The units of acceleration are \(\mathrm{mi} / \mathrm{hr}^{2}\).

Let \(R\) be the region bounded by the following curves. Use the disk or washer method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=x, y=2 x, y=6$$

Use any method (including geometry) to find the area of the following regions. In each case, sketch the bounding curves and the region in question. The region bounded by \(y=e^{x}, y=2 e^{-x}+1,\) and \(x=0\)

Evaluate the derivatives of the following functions. $$h(t)=(\sin t)^{\sqrt{t}}$$

Bankers use the law of \(70,\) which says that if an account increases at a fixed rate of \(p \% /\) yr, its doubling time is approximately 70/p. Explain why and when this statement is true.