Chapter 5

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ g(x)=\frac{4 \ln x}{x}, \quad y=0, \quad x=5 $$

The region bounded by \(y=\sqrt{x}, y=0, x=0,\) and \(x=4\) is revolved about the \(x\) -axis. (a) Find the value of \(x\) in the interval [0,4] that divides the solid into two parts of equal volume. (b) Find the values of \(x\) in the interval [0,4] that divide the solid into three parts of equal volume.

The level of sound \(\beta\) (in decibels) with an intensity of \(I\) is $$\beta(I)=10 \log _{10} \frac{I}{I_{0}}$$ where \(I_{0}\) is an intensity of \(10^{-16}\) watt per square centimeter, corresponding roughly to the faintest sound that can be heard. Determine \(\beta(I)\) for the following. (a) \(I=10^{-14}\) watt per square centimeter (whisper) (b) \(I=10^{-9}\) watt per square centimeter (busy street corner) (c) \(I=10^{-6.5}\) watt per square centimeter (air hammer) (d) \(I=10^{-4}\) watt per square centimeter (threshold of pain)

In Exercises \(41-44\), set up and evaluate the integrals for finding the area and moments about the \(x\) - and y-axes for the region bounded by the graphs of the equations. (Assume \(\rho=1\).) $$ y=x^{2}-4, y=0 $$

Volume of a Torus } Repeat Exercise 43 for a torus formed by revolving the region bounded by the circle \(x^{2}+y^{2}=r^{2}\) about the line \(x=R,\) where \(r

In Exercises \(45-48,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=\sqrt{\frac{x^{3}}{4-x}}, y=0, x=3 $$

The value of a tract of timber is\(V(t)=100,000 e^{0.8 \sqrt{t}}\) where \(t\) is the time in years, with \(t=0\) corresponding to 1998 . If money earns interest continuously at \(10 \%,\) the present value of the timber at any time \(t\) is \(A(t)=V(t) e^{-0.10 t} .\) Find the year in which the timber should be harvested to maximize the present value function.

What precalculus formula and representative element are used to develop the integration formula for the area of a surface of revolution?

In Exercises \(45-48\), use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. $$ y=10 x \sqrt{125-x^{3}}, y=0 $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=\sqrt{x} e^{x}, \quad y=0, x=0, x=1 $$

On the Richter scale, the magnitude \(R\) of an earthquake of intensity \(I\) is \(R=\frac{\ln I-\ln I_{0}}{\ln 10}\) where \(I_{0}\) is the minimum intensity used for comparison. Assume that \(I_{0}=1\) (a) Find the intensity of the 1906 San Francisco earthquake \((R=8.3)\) (b) Find the factor by which the intensity is increased if the Richter scale measurement is doubled. (c) Find \(d R / d I\).

In Exercises \(45-48\), use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. $$ y=x e^{-x / 2}, y=0, x=0, x=4 $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=x^{2}, \quad y=4 \cos x $$

A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

A right circular cone is generated by revolving the region bounded by \(y=h x / r, y=h,\) and \(x=0\) about the \(y\) -axis. Verify that the lateral surface area of the cone is \(S=\pi r \sqrt{r^{2}+h^{2}}\)

In Exercises \(47-50\), the integral represents the volume of a solid of revolution. Identify (a) the plane region that is revolved and (b) the axis of revolution. $$ 2 \pi \int_{0}^{2} x^{3} d x $$

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) explain why the area of the region is difficult to find by hand, and (c) use the integration capabilities of the graphing utility to approximate the area to four decimal places. $$ y=x^{2}, \quad y=\sqrt{3+x} $$

Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cylinder (b) Ellipsoid (c) Sphere (d) Right circular cone (e) Torus (i) \(\pi \int_{0}^{h}\left(\frac{r x}{h}\right)^{2} d x\) (ii) \(\pi \int_{0}^{h} r^{2} d x\) (iii) \(\pi \int_{-r}^{r}\left(\sqrt{r^{2}-x^{2}}\right)^{2} d x\) (iv) \(\pi \int_{-b}^{b}\left(a \sqrt{1-\frac{x^{2}}{b^{2}}}\right)^{2} d x\) (v) \(\pi \int_{-r}^{r}\left[\left(R+\sqrt{r^{2}-x^{2}}\right)^{2}-\left(R-\sqrt{r^{2}-x^{2}}\right)^{2}\right] d x\)

A sphere of radius \(r\) is generated by revolving the graph of \(y=\sqrt{r^{2}-x^{2}}\) about the \(x\) -axis. Verify that the surface area of the sphere is \(4 \pi r^{2}\)

In Exercises \(45-48\), use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. Witch of Agnesi \(y=8 /\left(x^{2}+4\right), y=0, x=-2, x=2\)

The integral represents the volume of a solid of revolution. Identify (a) the plane region that is revolved and (b) the axis of revolution. $$ 2 \pi \int_{0}^{1}\left(y-y^{3 / 2}\right) d y $$

In Exercises \(49-52,\) find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(x)=\int_{0}^{x}\left(\frac{1}{2} t+1\right) d t \quad \text { (a) } F(0) \quad \text { (b) } F(2) \quad \text { (c) } F(6) $$

Find the area of the zone of a sphere formed by revolving the graph of \(y=\sqrt{9-x^{2}}, 0 \leq x \leq 2\), about the \(y\) -axis.

The integral represents the volume of a solid of revolution. Identify (a) the plane region that is revolved and (b) the axis of revolution. $$ 2 \pi \int_{0}^{6}(y+2) \sqrt{6-y} d y $$

Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(x)=\int_{0}^{x}\left(\frac{1}{2} t^{2}+2\right) d t \quad \text { (a) } F(0) \quad \text { (b) } F(4) \quad \text { (c) } F(6) $$

Find the area of the zone of a sphere formed by revolving the graph of \(y=\sqrt{r^{2}-x^{2}}, 0 \leq x \leq a,\) about the \(y\) -axis. Assume that \(a

Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(\alpha)=\int_{-1}^{\alpha} \cos \frac{\pi \theta}{2} d \theta \quad \text { (a) } F(-1) \quad \text { (b) } F(0) \quad \text { (c) } F\left(\frac{1}{2}\right) $$

Bulb Design An ornamental light bulb is designed by revolving the graph of \(y=\frac{1}{3} x^{1 / 2}-x^{3 / 2}, \quad 0 \leq x \leq \frac{1}{3}\) about the \(x\) -axis, where \(x\) and \(y\) are measured in feet (see figure). Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Assume that the glass is 0.015 inch thick.)

\mathrm{Volume of a Segment of a Sphere } Let a sphere of radius \(r\) be cut by a plane, thereby forming a segment of height \(h\). Show that the volume of this segment is \(\frac{1}{3} \pi h^{2}(3 r-h)\).

Find the accumulation function \(F\). Then evaluate \(F\) at each value of the independent variable and graphically show the area given by each value of \(F\). $$ F(y)=\int_{-1}^{y} 4 e^{x / 2} d x \quad \text { (a) } F(-1) \quad \text { (b) } F(0) \quad \text { (c) } F(4) $$

A manufacturer drills a hole through the center of a metal sphere of radius \(R\). The hole has a radius \(r\). Find the volume of the resulting ring.

Think About It Consider the equation \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1 .\) (a) Use a graphing utility to graph the equation. (b) Set up the definite integral for finding the first quadrant arc length of the graph in part (a). (c) Compare the interval of integration in part (b) and the domain of the integrand. Is it possible to evaluate the definite integral? Is it possible to use Simpson's Rule to evaluate the definite integral? Explain. (You will learn how to evaluate this type of integral in Section \(6.7 .)\)

Volume of an Ellipsoid Consider the plane region bounded by the graph of \(\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=1\) where \(a>0\) and \(b>0\). Show that the volume of the ellipsoid formed when this region revolves about the \(y\) -axis is \(\frac{4 \pi a^{2} b}{3}\).

In Exercises \(53-56,\) use integration to find the area of the figure having the given vertices. $$ (2,-3),(4,6),(6,1) $$

Let \(R\) be the region bounded by \(y=1 / x,\) the \(x\) -axis, \(x=1,\) and \(x=b,\) where \(b>1 .\) Let \(D\) be the solid formed when \(R\) is revolved about the \(x\) -axis. (a) Find the volume \(V\) of \(D\). (b) Write the surface area \(S\) as an integral. (c) Show that \(V\) approaches a finite limit as \(b \rightarrow \infty\). (d) Show that \(S \rightarrow \infty\) as \(b \rightarrow \infty\).

Think About It Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cone (b) Torus (c) Sphere (d) Right circular cylinder (e) Ellipsoid (i) \(2 \pi \int_{0}^{r} h x d x\) (ii) \(2 \pi \int_{0}^{r} h x\left(1-\frac{x}{r}\right) d x\) (iii) \(2 \pi \int_{0}^{r} 2 x \sqrt{r^{2}-x^{2}} d x\) (iv) \(2 \pi \int_{0}^{b} 2 a x \sqrt{1-\frac{x^{2}}{b^{2}}} d x\) (v) \(2 \pi \int_{-r}^{r}(R-x)\left(2 \sqrt{r^{2}-x^{2}}\right) d x\)

\mathrm{\\{} I n d i v i d u a l ~ P r o j e c t ~ \(\quad\) Select a solid of revolution from everyday life. Measure the radius of the solid at a minimum of seven points along its axis. Use the data to approximate the volume of the solid and the surface area of the lateral sides of the solid.

Volume of a Storage Shed A storage shed has a circular base of diameter 80 feet (see figure). Starting at the center, the interior height is measured every 10 feet and recorded in the table. \begin{tabular}{|l|c|c|c|c|c|} \hline\(x\) & 0 & 10 & 20 & 30 & 40 \\ \hline Height & 50 & 45 & 40 & 20 & 0 \\ \hline \end{tabular} (a) Use Simpson's Rule to approximate the volume of the shed. (b) Note that the roof line consists of two line segments. Find the equations of the line segments and use integration to find the volume of the shed.

Writing Read the article "Arc Length, Area and the Arcsine Function" by Andrew M. Rockett in Mathematics Magazine. Then write a paragraph explaining how the arcsine function can be defined in terms of an arc length. (To view this article, go to the website www.matharticles.com.)

Astroid Find the area of the surface formed by revolving the portion in the first quadrant of the graph of \(x^{2 / 3}+y^{2 / 3}=4\) \(0 \leq y \leq 8\) about the \(y\) -axis.

In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \((x-5)^{2}+y^{2}=16\) about the \(y\) -axis

In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle \(x^{2}+(y-3)^{2}=4\) about the \(x\) -axis.

Let \(V_{1}\) and \(V_{2}\) be the volumes of the solids that result when the plane region bounded by \(y=1 / x, y=0, x=\frac{1}{4},\) and \(x=c\left(c>\frac{1}{4}\right)\) is revolved about the \(x\) -axis and \(y\) -axis, respectively. Find the value of \(c\) for which \(V_{1}=V_{2}\)

In Exercises 59 and 60 , set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ f(x)=\frac{1}{x^{2}+1}, \quad\left(1, \frac{1}{2}\right) $$

Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2} .\) Let \(h\) represent the height of the cable from its lowest point to its highest point and let \(2 w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by \(C=2 \int_{0}^{w} \sqrt{1+\frac{4 h^{2}}{w^{4}} x^{2}} d x\)

In Exercises \(57-60\), use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of \(y=x, y=4,\) and \(x=0\) about the \(x\) -axis

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point. $$ y=x^{3}-2 x, \quad(-1,1) $$

The solid formed by revolving the region bounded by the graphs of \(y=x, y=4,\) and \(x=0\) about the \(x\) -axis The solid formed by revolving the region bounded by the graphs of \(y=2 \sqrt{x-2}, y=0,\) and \(x=6\) about the \(y\) -axis

The graphs of \(y=x^{4}-2 x^{2}+1\) and \(y=1-x^{2}\) intersect at three points. However, the area between the curves can be found by a single integral. Explain why this is so, and write an integral for this area.

Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.