Chapter 6

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

Bernoulli's “parabolas" Johann Bernoulli (1667-1748) evaluated the arc length of curves of the form \(y=x^{(2 n+1) / 2 n},\) where \(n\) is a positive integer, on the interval \([0, a].\) a. Write the arc length integral. b. Make the change of variables \(u^{2}=1+\left(\frac{2 n+1}{2 n}\right)^{2} x^{1 / n}\) to obtain a new integral with respect to \(u.\) c. Use the Binomial Theorem to expand this integrand and evaluate the integral. d. The case \(n=1\left(y=x^{3 / 2}\right)\) was done in Example \(1 .\) With \(a=1,\) compute the arc length in the cases \(n=2\) and \(n=3\) Does the arc length increase or decrease with \(n\) ? e. Graph the arc length of the curves for \(a=1\) as a function of \(n.\)

Show that the surface area of the frustum of a cone generated by revolving the line segment between \((a, g(a))\) and \((b, g(b))\) about the \(x\) -axis is \(\pi(g(b)+g(a)) \ell,\) for any linear function \(g(x)=c x+d\) that is positive on the interval \([a, b],\) where \(\ell\) is the slant height of the frustum.

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. $$y=0, y=\ln x, y=2, x=0$$

The owners of an oil reserve begin extracting oil at time \(t=0 .\) Based on estimates of the reserves, suppose the projected extraction rate is given by \(Q^{\prime}(t)=3 t^{2}(40-t)^{2}\) where \(0 \leq t \leq 40, Q\) is measured in millions of barrels, and \(t\) is measured in years. a. When does the peak extraction rate occur? b. How much oil is extracted in the first \(10,20,\) and 30 years? c. What is the total amount of oil extracted in 40 years? d. Is one-fourth of the total oil extracted in the first one-fourth of the extraction period? Explain.

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 below the line \(y=2\) and above the curve \(y=\sec ^{2} x\) on the interval \([0, \pi / 4]\)

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. $$y=2$$

An inverted cone is \(2 \mathrm{m}\) high and has a base radius of \(\frac{1}{2} \mathrm{m}\). If the tank is full, how much work is required to pump the water to a level \(1 \mathrm{m}\) above the top of the tank?

Evaluate the derivatives of the following functions. $$H(x)=(x+1)^{2 x}$$

The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthly and annually. Suppose that, for a particular month, the monthly rate of inflation is reported as \(0.8 \%\). Assuming that this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.

Evaluate each definite integral. \(\int_{0}^{1} \cosh ^{3} 3 x \sinh 3 x d x\)

Let \(f\) be differentiable on \([a, b]\) and suppose that \(g(x)=c f(x)\) and \(h(x)=f(c x),\) where \(c>0 .\) When the curve \(y=f(x)\) on \([a, b]\) is revolved about the \(x\) -axis, the area of the resulting surface is \(A\). Evaluate the following integrals in terms of \(A\) and \(c\). a. \(\int_{a}^{b} g(x) \sqrt{c^{2}+g^{\prime}(x)^{2}} d x\) b. \(\int_{a / c}^{b / c} h(x) \sqrt{c^{2}+h^{\prime}(x)^{2}} d x\)

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. $$y=x^{3}, y=0, x=2$$

An oil refinery produces oil at a variable rate given by $$Q^{\prime}(t)=\left\\{\begin{array}{ll}800 & \text { if } 0 \leq t<30 \\\2600-60 t & \text { if } 30 \leq t<40 \\\200 & \text { if } t \geq 40,\end{array}\right.$$ where \(t\) is measured in days and \(Q\) is measured in barrels. a. How many barrels are produced in the first 35 days? b. How many barrels are produced in the first 50 days? c. Without using integration, determine the number of barrels produced over the interval [60,80].

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 between the line \(y=x\) and the curve \(y=2 x \sqrt{1-x^{2}}\) in the first quadrant

Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved about the following lines. $$y=-2$$

Evaluate the derivatives of the following functions. $$p(x)=x^{-\ln x}$$

Suppose the acceleration of an object moving along a line is given by \(a(t)=-k v(t),\) where \(k\) is a positive constant and \(v\) is the object's velocity. Assume that the initial velocity and position are given by \(v(0)=10\) and \(s(0)=0,\) respectively. a. Use \(a(t)=v^{\prime}(t)\) to find the velocity of the object as a function of time. b. Use \(v(t)=s^{\prime}(t)\) to find the position of the object as a function of time. c. Use the fact that \(d v / d t=(d v / d s)(d s / d t)\) (by the Chain Rule) to find the velocity as a function of position.

Evaluate each definite integral. \(\int_{0}^{4} \frac{\operatorname{sech}^{2} \sqrt{x}}{\sqrt{x}} d x\)

Starting with an initial value of \(P(0)=55,\) the population of a prairie dog community grows at a rate of \(P^{\prime}(t)=20-t / 5\) (in units of prairie dogs/month), for \(0 \leq t \leq 200\). a. What is the population 6 months later? b. Find the population \(P(t),\) for \(0 \leq t \leq 200\).

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. $$y=\sqrt{x}, y=0, x=4$$

Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved about the following lines. $$x=-1$$

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 \(x=y^{2}-4\) and \(y=x / 3\)

Evaluate each definite integral. \(\int_{0}^{\ln 2} \tanh x d x\)

When records were first kept \((t=0),\) the population of a rural town was 250 people. During the following years, the population grew at a rate of \(P^{\prime}(t)=30(1+\sqrt{t}),\) where \(t\) is measured in years. a. What is the population after 20 years? b. Find the population \(P(t)\) at any time \(t \geq 0\).

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. $$x=\sqrt{4-y^{2}}, x=0$$

Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by \(y=x\) and \(x=y^{2}\) can be found only by integrating with respect to \(x\) b. The area of the region between \(y=\sin x\) and \(y=\cos x\) on the interval \([0, \pi / 2]\) is \(\int_{0}^{\pi / 2}(\cos x-\sin x) d x\) c. \(\int_{0}^{1}\left(x-x^{2}\right) d x=\int_{0}^{1}(\sqrt{y}-y) d y\)

Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved about the following lines. $$y=6$$

Evaluate each definite integral. \(\int_{\ln 2}^{\ln 3} \operatorname{csch} x d x\)

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. $$y=\sin ^{-1} x, x=0, y=\pi / 4$$

Sketch the region and find its area. The region bounded by \(y=\sin x\) and \(y=x(x-\pi),\) for \(0 \leq x \leq \pi\)

A model for the startup of a runner in a short race results in the velocity function \(v(t)=a\left(1-e^{-t / c}\right),\) where \(a\) and \(c\) are positive constants and \(v\) has units of \(\mathrm{m} / \mathrm{s}\). (Source: A Theory of Competitive Running, Joe Keller, Physics Today 26 (September 1973)) a. Graph the velocity function for \(a=12\) and \(c=2 .\) What is the runner's maximum velocity? b. Using the velocity in part (a) and assuming \(s(0)=0\), find the position function \(s(t),\) for \(t \geq 0\) c. Graph the position function and estimate the time required to run \(100 \mathrm{m} .\)

Use either the washer or shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by \(y=x^{2}, y=4,\) and \(x=0\) is revolved about the following lines. $$x=2$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume \(x>0\) and \(y>0\) a. \(\ln x y=\ln x+\ln y\) b. \(\ln 0=1\) c. \(\ln (x+y)=\ln x+\ln y\) d. \(2^{x}=e^{2 \ln x}\) e. The area under the curve \(y=1 / x\) and the \(x\) -axis on the interval \([1, e]\) is 1

Suppose the cells of a tumor are idealized as spheres each with a radius of \(5 \mu \mathrm{m}\) (micrometers). The number of cells has a doubling time of 35 days. Approximately how long will it take a single cell to grow into a multi- celled spherical tumor with a volume of \(0.5 \mathrm{cm}^{3}(1 \mathrm{cm}=10,000 \mu \mathrm{m}) ?\) Assume that the tumor spheres are tightly packed.

Evaluate the following integrals two ways. a. Simplify the integrand first, and then integrate. b. Change variables (let \(u=\ln x\) ), integrate, and then simplify your answer. Verify that both methods give the same answer. \(\int \frac{\sinh (\ln x)}{x} d x\)

A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of \(N^{\prime}(t)=100 e^{-0.25 t}\). a. What is the population after 20 days? After 40 days? b. Find the population \(N(t)\) at any time \(t \geq 0\).

For the following regions \(R\), determine which is greater - the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or about the y-axis. \(R\) is bounded by \(y=2 x,\) the \(x\) -axis, and \(x=5\)

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=x, y=x^{1 / 3} ;\) in the first quadrant; revolved about the \(x\) -axis

Sketch the region and find its area. The region bounded by \(y=(x-1)^{2}\) and \(y=7 x-19\)

Logarithm properties Use the integral definition of the natural logarithm to prove that \(\ln (x / y)=\ln x-\ln y\)

The burning of fossil fuels releases greenhouse gases into the atmosphere. In \(1995,\) the United States emitted about 1.4 billion tons of carbon into the atmosphere, nearly one-fourth of the world total. China was the second largest contributor, emitting about 850 million tons of carbon. However, emissions from China were rising at a rate of about \(4 \% / \mathrm{yr},\) while U.S. emissions were rising at about \(1.3 \% /\) yr. Using these growth rates, project greenhouse gas emissions from the United States and China in \(2020 .\) Graph the projected emissions for both countries. Comment on your observations.

Evaluate the following integrals two ways. a. Simplify the integrand first, and then integrate. b. Change variables (let \(u=\ln x\) ), integrate, and then simplify your answer. Verify that both methods give the same answer. \(\int_{1}^{\sqrt{3}} \frac{\operatorname{sech}(\ln x)}{x} d x\)

For the following regions \(R\), determine which is greater - the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or about the y-axis. \(R\) is bounded by \(y=4-2 x,\) the \(x\) -axis, and the \(y\) -axis.

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=x^{2} / 8, y=2-x,\) and \(x=0 ;\) revolved about the \(y\) -axis

Sketch the region and find its area. The region bounded by \(y=2\) and \(y=\frac{1}{\sqrt{1-x^{2}}}\)

A large building shaped like a box is 50 \(\mathrm{m}\) high with a face that is \(80 \mathrm{m}\) wide. A strong wind blows directly at the face of the building, exerting a pressure of \(150 \mathrm{N} / \mathrm{m}^{2}\) at the ground and increasing with height according to \(P(y)=150+2 y,\) where \(y\) is the height above the ground. Calculate the total force on the building, which is a measure of the resistance that must be included in the design of the building.

The owner of a clothing store understands that the demand for shirts decreases with the price. In fact, she has developed a model that predicts that at a price of \(\$ x\) per shirt, she can sell \(D(x)=40 e^{-x / 50}\) shirts in a day. It follows that the revenue (total money taken in) in a day is \(R(x)=x D(x)(\$ x / \text { shirt } \cdot D(x) \text { shirts }) .\) What price should the owner charge to maximize revenue?

a. Use a graphing utility to sketch the graph of \(y=\operatorname{coth} x\), and then explain why \(\int_{5}^{10} \operatorname{coth} x d x \approx 5\). b. Evaluate \(\int_{5}^{10} \operatorname{coth} x d x\) analytically and use a calculator to arrive at a decimal approximation to the answer. How large is the error in the approximation in part (a)?

For the following regions \(R\), determine which is greater - the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or about the y-axis. \(R\) is bounded by \(y=1-x^{3}\), the \(x\) -axis, and the \(y\) -axis.