Chapter 6

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves in Exercises \(19-28\) about the \(x\) -axis. \begin{equation} y=x^{2}, \quad y=0, \quad x=2 \end{equation}

In Exercises \(13-20,\) do the following. a. Set up an integral for the length of the curve. b. Graph the curve to see what it looks like. c. Use your grapher's or computer's integral evaluator to find the curve's length numerically. $$x=\int_{0}^{y} \sqrt{\sec ^{2} t-1} d t, \quad-\pi / 3 \leq y \leq \pi / 4$$

A right-circular cylindrical tank of height 10 \(\mathrm{ft}\) and radius 5 \(\mathrm{ft}\) is lying horizontally and is full of diesel fuel weighing 53 \(\mathrm{lb/ft}^{3}\) . How much work is required to pump all of the fuel to a point 15 \(\mathrm{ft}\) above the top of the tank?

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=x^{3}, \quad y=0, \quad x=2 \end{equation}

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=\sqrt{9-x^{2}}, \quad y=0 \end{equation}

Find the areas of the surfaces generated by revolving the curves in Exercises \(13-23\) about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. \(y=(1 / 3)\left(x^{2}+2\right)^{3 / 2}, \quad 0 \leq x \leq \sqrt{2} ; \quad y\) -axis (Hint: Express \(d s=\sqrt{d x^{2}+d y^{2}}\) in terms of \(d x,\) and evaluate the integral \(S=\int 2 \pi x d s\) with appropriate limits.)

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=x-x^{2}, \quad y=0 \end{equation}

Find the length of the curve $$y=\int_{0}^{x} \sqrt{\cos 2 t} d t$$ from \(x=0\) to \(x=\pi / 4\)

Kinetic energy If a variable force of magnitude \(F(x)\) moves an object of mass \(m\) along the \(x\) -axis from \(x_{1}\) to \(x_{2}\) , the object's velocity \(v\) can be written as \(d x / d t\) (where \(t\) represents time). Use Newton's second law of motion \(F=m(d v / d t)\) and the Chain Rule $$\frac{d v}{d t}=\frac{d v d x}{d x d t}=v \frac{d v}{d x}$$ to show that the net work done by the force in moving the object from \(x_{1}\) to \(x_{2}\) is $$W=\int_{x_{1}}^{x_{2}} F(x) d x=\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}$$ where \(v_{1}\) and \(v_{2}\) are the object's velocities at \(x_{1}\) and \(x_{2} .\) In physics, the expression \((1 / 2) m v^{2}\) is called the kinetic energy of an object of mass \(m\) moving with velocity \(v\) . Therefore, the work done by the force equals the change in the object's kinetic energy, and we can find the work by calculating this change.

In Exercises \(23-26,\) use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. $$y=3 x, \quad y=0, \quad x=2$$ $$\begin{array}{ll}{\text { a. The } y \text { -axis }} & {\text { b. The line } x=4} \\ {\text { c. The line } x=-1} & {\text { d. The } x \text { -axis }} \\\ {\text { e. The line } y=7} & {\text { f. The line } y=-2}\end{array}$$

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=\sqrt{\cos x}, \quad 0 \leq x \leq \pi / 2, \quad y=0, \quad x=0 \end{equation}

The length of an astroid The graph of the equation \(x^{2 / 3}+\) \(y^{2 / 3}=1\) is one of a family of curves called astroids (not "asteroids") because of their star like appearance (see the accompanying figure). Find the length of this particular astroid by finding the length of half the first-quadrant portion, \(y=\left(1-x^{2 / 3}\right)^{3 / 2}\) \(\sqrt{2} / 4 \leq x \leq 1,\) and multiplying by \(8 .\)

Tennis \(A\) -oz tennis ball was served at 160 \(\mathrm{ft} / \mathrm{sec}\) (about 109 \(\mathrm{mph}\) ). How much work was done on the ball to make it go this fast? (To find the ball's mass from its weight, express the weight in pounds and divide by \(32 \mathrm{ft} / \mathrm{sec}^{2},\) the acceleration of gravity.)

Write an integral for the area of the surface generated by revolving the curve \(y=\cos x,-\pi / 2 \leq x \leq \pi / 2,\) about the \(x\) -axis. In Section 8.4 we will see how to evaluate such integrals.

In Exercises \(23-26,\) use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. $$y=x^{3}, \quad y=8, \quad x=0$$ $$\begin{array}{ll}{\text { a. The } y \text { -axis }} & {\text { b. The line } x=3} \\ {\text { c. The line } x=-2} & {\text { d. The } x \text { -axis }} \\\ {\text { e. The line } y=8} & {\text { f. The line } y=-1}\end{array}$$

Find the volumes of the solids generated by revolving the regions bounded by the lines and curves about the \(x\) -axis. \begin{equation} y=\sec x, \quad y=0, \quad x=-\pi / 4, \quad x=\pi / 4 \end{equation}

Testing the new definition Show that the surface area of a sphere of radius \(a\) is still 4\(\pi a^{2}\) by using Equation \((3)\) to find the area of the surface generated by revolving the curve \(y=\sqrt{a^{2}-x^{2}},-a \leq x \leq a,\) about the \(x\) -axis.

In Exercises \(23-26,\) use the shell method to find the volumes of the solids generated by revolving the regions bounded by the given curves about the given lines. $$y=x+2, \quad y=x^{2}$$ $$\begin{array}{ll}{\text { a. The line } x=2} & {\text { b. The line } x=-1} \\\ {\text { c. The } x \text { -axis }} & {\text { d. The line } y=4}\end{array}$$

In Exercises 25 and \(26,\) find the volume of the solid generated by revolving the region about the given line. \begin{equation} \begin{array}{l}{\text { The region in the first quadrant bounded above by the line }} \\ {y=\sqrt{2}, \text { below by the curve } y=\sec x \tan x, \text { and on the left by }} \\ {\text { the } y \text { -axis, about the line } y=\sqrt{2}}\end{array} \end{equation}

Testing the new definition The lateral (side) surface area of a cone of height \(h\) and base radius \(r\) should be \(\pi r \sqrt{r^{2}+h^{2}}\) , the semiperimeter of the base times the slant height. Show that this is still the case by finding the area of the surface generated by revolving the line segment \(y=(r / h) x, 0 \leq x \leq h,\) about the \(x\) -axis.

Find the volume of the solid generated by revolving the region about the given line. \begin{equation} \begin{array}{l}{\text { The region in the first quadrant bounded above by the line } y=2} \\ {\text { below by the curve } y=2 \sin x, 0 \leq x \leq \pi / 2, \text { and on the left by }} \\ {\text { the } y \text { -axis, about the line } y=2}\end{array} \end{equation}

\begin{equation} \begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array} \end{equation} The region enclosed by \(x=\sqrt{5} y^{2}, \quad x=0, \quad y=-1, \quad y=1\)

Is there a smooth (continuously differentiable) curve \(y=f(x)\) whose length over the interval \(0 \leq x \leq a\) is always \(\sqrt{2} a ?\) Give reasons for your answer.

For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to \(y .\) It may not be possible, however, to express the integrand in terms of \(y .\) In such a case, the shell method allows us to integrate with respect to \(x\) instead. Exercises 29 and 30 provide some insight. Compute the volume of the solid generated by revolving the region bounded by \(y=x\) and \(y=x^{2}\) about each coordinate axis using a. the shell method. \(\quad\) b. the washer method.

\begin{equation}\begin{array}{l}{\text { Find the volumes of the solids generated by revolving the regions }} \\ {\text { bounded by the lines and curves in Exercises } 27-32 \text { about the } y \text { -axis. }}\end{array}\end{equation} The region enclosed by \(x=\sqrt{2 \sin 2 y}, \quad 0 \leq y \leq \pi / 2, \quad x=0\)

For some regions, both the washer and shell methods work well for the solid generated by revolving the region about the coordinate axes, but this is not always the case. When a region is revolved about the y-axis, for example, and washers are used, we must integrate with respect to \(y .\) It may not be possible, however, to express the integrand in terms of \(y .\) In such a case, the shell method allows us to integrate with respect to \(x\) instead. Exercises 29 and 30 provide some insight. Compute the volume of the solid generated by revolving the triangular region bounded by the lines \(2 y=x+4, y=x,\) and \(x=0\) about $$ \begin{array}{l}{\text { a. the } x \text { -axis using the washer method. }} \\\ {\text { b. the } y \text { -axis using the shell method. }} \\ {\text { c. the line } x=4 \text { using the shell method. }} \\ {\text { d. the line } y=8 \text { using the washer method. }}\end{array} $$

Putting a satellite in orbit The strength of Earth's gravitational field varies with the distance \(r\) from Earth's center, and the magnitude of the gravitational force experienced by a satellite of mass \(m\) during and after launch is $$F(r)=\frac{m M G}{r^{2}}$$ Here, \(M=5.975 \times 10^{24} \mathrm{kg}\) is Earth's mass, \(G=6.6720 \times\) \(10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} \mathrm{kg}^{-2}\) is the universal gravitational constant, and \(r\) is measured in meters. The work it takes to lift a \(1000-\mathrm{kg}\) satellite from Earth's surface to a circular orbit \(35,780 \mathrm{km}\) above Earth's center is therefore given by the integral \begin{equation} =\int_{6,370,000}^{35,780,000} \frac{1000 M G}{r^{2}} d r \ joules. \end{equation} Evaluate the integral. The lower limit of integration is Earth's radius in meters at the launch site. (This calculation does not take into account energy spent lifting the launch vehicle or energy spent bringing the satellite to orbit velocity.)

In Exercises \(31-36,\) find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The triangle with vertices \((1,1),(1,2),\) and \((2,2)\) about $$\begin{array}{ll}{\text { a. the } x \text { -axis }} & {\text { b. the } y \text { -axis }} \\ {\text { c. the line } x=10 / 3} & {\text { d. the line } y=1}\end{array}$$

Forcing electrons together each other with a force of $$F=\frac{23 \times 10^{-29}}{r^{2}}$$ \begin{equation} \begin{array}{l}{\text { a. Suppose one electron is held fixed at the point }(1,0) \text { on the }} \\ {x \text { -axis (units in meters). How much work does it take to move }} \\ {\text { a second electron along the } x \text { -axis from the point }(-1,0) \text { to }} \\ {\text { the origin? }}\\\\{\text { b. Suppose an electron is held fixed at each of the points }(-1,0)} \\ {\text { and }(1,0) . \text { How much work does it take to move a third elec- }} \\ {\text { tron along the } x \text { -axis from }(5,0) \text { to }(3,0) ?}\end{array} \end{equation}

In Exercises \(31-36,\) find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The region bounded by \(y=\sqrt{x}, y=2, x=0\) about $$\begin{array}{ll}{\text { a. the } x \text { -axis }} & {\text { b. the } y \text { -axis }} \\ {\text { c. the line } x=4} & {\text { d. the line } y=2}\end{array}$$

Find the arc length function for the graph of \(f(x)=2 x^{3 / 2}\) using \((0,0)\) as the starting point. What is the length of the curve from \((0,0)\) to \((1,2) ?\)

In Exercises \(31-36,\) find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The region in the first quadrant bounded by \(x=y-y^{3}, x=1\) and \(y=1\) about $$\begin{array}{ll}{\text { a. the } x \text { -axis }} & {\text { b. the } y \text { -axis }} \\ {\text { c. the line } x=1} & {\text { d. the line } y=1}\end{array}$$

In Exercises \(35-40\) , use a CAS to perform the following steps for the given graph of the function over the closed interval. a. Plot the curve together with the polygonal path approximations for \(n=2,4,8\) partition points over the interval. Figure \(6.22 . )\) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for \(n=2,4,8\) with the actual length given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$f(x)=\sqrt{1-x^{2}},-1 \leq x \leq 1$$

In Exercises \(31-36,\) find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The region bounded by \(y=\sqrt{x}\) and \(y=x^{2} / 8\) about a. the \(x\) -axis \(\quad\) b. the \(y\) -axis

In Exercises \(35-40\) , use a CAS to perform the following steps for the given graph of the function over the closed interval. a. Plot the curve together with the polygonal path approximations for \(n=2,4,8\) partition points over the interval. Figure \(6.22 . )\) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for \(n=2,4,8\) with the actual length given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$f(x)=x^{1 / 3}+x^{2 / 3}, \quad 0 \leq x \leq 2$$

Semicircular plate Calculate the fluid force on one side of a semicircular plate of radius 5 \(\mathrm{ft}\) that rests vertically on its diameter at the bottom of a pool filled with water to a depth of 6 \(\mathrm{ft}\) .

In Exercises \(31-36,\) find the volumes of the solids generated by revolving the regions about the given axes. If you think it would be better to use washers in any given instance, feel free to do so. The region bounded by \(y=2 x-x^{2}\) and \(y=x\) about a. the \(y\) -axis \(\quad\) b. the line \(x=1\)

In Exercises \(35-40\) , use a CAS to perform the following steps for the given graph of the function over the closed interval. a. Plot the curve together with the polygonal path approximations for \(n=2,4,8\) partition points over the interval. Figure \(6.22 . )\) b. Find the corresponding approximation to the length of the curve by summing the lengths of the line segments. c. Evaluate the length of the curve using an integral. Compare your approximations for \(n=2,4,8\) with the actual length given by the integral. How does the actual length compare with the approximations as \(n\) increases? Explain your answer. $$f(x)=\sin \left(\pi x^{2}\right), \quad 0 \leq x \leq \sqrt{2}$$

The region in the first quadrant that is bounded above by the curve \(y=1 / x^{1 / 4}\) , on the left by the line \(x=1 / 16,\) and below by the line \(y=1\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid by a. the washer method. \(\quad\) b. the shell method.

The region in the first quadrant that is bounded above by the curve \(y=1 / \sqrt{x},\) on the left by the line \(x=1 / 4,\) and below by the line \(y=1\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid by a. the washer method. \(\quad\) b. the shell method.

New England Aquarium The viewing portion of the rectangular glass window in a typical fish tank at the New England Aquarium in Boston is 63 in. wide and runs from 0.5 in. below the water's surface to 33.5 in. below the surface. Find the fluid force against this portion of the window. The weight-density of seawater is 64 \(\mathrm{lb} / \mathrm{ft}^{3} .\) (In case you were wondering, the glass is 3\(/ 4\) in. thick and the tank walls extend 4 in. above the water to keep the fish from jumping out.)

Semicircular plate A semicircular plate 2 ft in diameter sticks straight down into freshwater with the diameter along the surface. Find the force exerted by the water on one side of the plate.

Tilted plate Calculate the fluid force on one side of a 5 ft by 5 ft square plate if the plate is at the bottom of a pool filled with water to a depth of 8 \(\mathrm{ft}\) and \begin{equation} \begin{array}{l}{\text { a. lying flat on its } 5 \mathrm{ft} \text { by } 5 \text { ft face. }} \\ {\text { b. resting vertically on a } 5 \text { -ft edge. }} \\ {\text { c. resting on a } 5 \text { -ft edge and tilted at } 45^{\circ} \text { to the bottom of the pool. }}\end{array} \end{equation}

A bead is formed from a sphere of radius 5 by drilling through a diameter of the sphere with a drill bit of radius \(3 .\) a. Find the volume of the bead. b. Find the volume of the removed portion of the sphere.

In Exercises \(41-44,\) find the volume of the solid generated by revolving each region about the \(y\) -axis. \begin{equation} \begin{array}{l}{\text { The region enclosed by the triangle with vertices }(1,0),(2,1),} \\ {\text { and }(1,1)}\end{array} \end{equation}

Tilted plate Calculate the fluid force on one side of a right- triangular plate with edges \(3 \mathrm{ft}, 4 \mathrm{ft}\) , and 5 \(\mathrm{ft}\) if the plate sits at the bottom of a pool filled with water to a depth of 6 \(\mathrm{ft}\) on its 3 -ft edge and tilted at \(60^{\circ}\) to the bottom of the pool.

A Bundt cake, well known for having a ringed shape, is formed by revolving around the \(y\) -axis the region bounded by the graph of \(y=\sin \left(x^{2}-1\right)\) and the \(x\) -axis over the interval \(1 \leq x \leq\) \(\sqrt{1+\pi} .\) Find the volume of the cake.

Find the volume of the solid generated by revolving each region about the \(y\) -axis. \begin{equation} \begin{array}{l}{\text { The region enclosed by the triangle with vertices }(0,1),(1,0),} \\ {\text { and }(1,1)}\end{array} \end{equation}

Derive the formula for the volume of a right circular cone of height \(h\) and radius \(r\) using an appropriate solid of revolution.

Find the volume of the solid generated by revolving each region about the \(y\) -axis. \begin{equation} \begin{array}{l}{\text { The region in the first quadrant bounded above by the parabola }} \\ {y=x^{2}, \text { below by the } x \text { -axis, and on the right by the line } x=2}\end{array} \end{equation}