Chapter 6

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the parabola \(x=y^{2}-y\) and the line \(y=x\)

Find the volume of the given pyramid, which has a square base of area 9 and height 5.

Find the lengths of the curves in Exercises \(1-12 .\) If you have graphing software, you may want to graph these curves to see what they look like. $$y=\int_{-2}^{x} \sqrt{3 t^{4}-1} d t, \quad-2 \leq x \leq-1$$

Force of attraction When a particle of mass \(m\) is at \((x, 0),\) it is attracted toward the origin with a force whose magnitude is \(k / x^{2}\) . If the particle starts from rest at \(x=b\) and is acted on by no other forces, find the work done on it by the time it reaches \(x=a\) \(0 < a < b .\)

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the \(x\) -axis and the curve \(y=\cos x,\) \(-\pi / 2 \leq x \leq \pi / 2\)

A twisted solid A square of side length \(s\) lies in a plane perpendicular to a line \(L .\) One vertex of the square lies on \(L .\) As this square moves a distance \(h\) along \(L,\) the square turns one revolution about \(L\) to generate a corkscrew-like column with square cross-sections. a. Find the volume of the column. b. What will the volume be if the square turns twice instead of once? Give reasons for your answer.

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$y=x^{2},-1 \leq x \leq 2$$

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. $$y = \sqrt { 2 x - x ^ { 2 } } , \quad 0.5 \leq x \leq 1.5 ; \quad x -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region between the curve \(y=\sec ^{2} x,-\pi / 4 \leq x \leq \pi / 4\) and the \(x\) -axis

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$y=\tan x, \quad-\pi / 3 \leq x \leq 0$$

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. $$y = \sqrt { 2 x - x ^ { 2 } } , \quad 0.5 \leq x \leq 1.5 ; \quad x -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. \begin{equation}\begin{array}{l}{\text { a. The region cut from the first quadrant by the circle } x^{2}+y^{2}=9} \\ {\text { b. The region bounded by the } x \text { -axis and the semicircle }} \\ {y=\sqrt{9-x^{2}}} \\ {\text { Compare your answer in part (b) with the answer in part (a). }}\end{array} \end{equation}

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$x=\sin y, \quad 0 \leq y \leq \pi$$

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. $$y = \sqrt { x + 1 } , \quad 1 \leq x \leq 5 ; \quad x -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded by the parabolas \(y=2 x^{2}-4 x\) and \(y=2 x-x^{2}\)

Gasoline in a tank A gasoline tank is in the shape of a right circular cylinder (lying on its side) of length 10 \(\mathrm{ft}\) and radius 4 ft. Set up an integral that represents the volume of the gas in the tank if it is filled to a depth of 6 ft. You will learn how to compute this integral in Chapter 8 (or you may use geometry to find its value).

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$x=\sqrt{1-y^{2}}, \quad-1 / 2 \leq y \leq 1 / 2$$

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. $$x = y ^ { 3 } / 3 , \quad 0 \leq y \leq 1 ; \quad y -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region between the curve \(y=1 / \sqrt{x}\) and the \(x\) -axis from \(x=1\) to \(x=16\)

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The region bounded above by the curve \(y=1 / x^{3}\) , below by the curve \(y=-1 / x^{3},\) and on the left and right by the lines \(x=1\) and \(x=a>1 .\) Also, find \(\lim _{a \rightarrow \infty} \overline{x}\)

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$y=\sin x-x \cos x, \quad 0 \leq x \leq \pi$$

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. $$x = 2 \sqrt { 4 - y } , \quad 0 \leq y \leq 15 / 4 ; \quad y -axis$$

Emptying a tank A vertical right-circular cylindrical tank measures 30 \(\mathrm{ft}\) high and 20 \(\mathrm{ft}\) in diameter. It is full of kerosene weighing 51.2 \(\mathrm{lb} / \mathrm{ft}^{3} .\) How much work does it take to pump the kerosene to the level of the top of the tank?

In Exercises \(13-20,\) do the following. \begin{equation}\begin{array}{l}{\text { a. Set up an integral for the length of the curve. }} \\ {\text { b. Graph the curve to see what it looks like. }} \\\ {\text { c. Use your grapher's or computer's integral evaluator to find }} \\\ {\text { the curve's length numerically. }}\end{array}\end{equation} $$y=\int_{0}^{x} \tan t d t, \quad 0 \leq x \leq \pi / 6$$

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. $$x = \sqrt { 2 y - 1 } , \quad 5 / 8 \leq y \leq 1 ; \quad y -axis$$

In Exercises \(18,\) find the center of mass of a thin plate of constant density \(\delta\) covering the given region. Consider a thin plate of constant density \(\delta\) lies in the region bounded by the graphs of \(y=\sqrt{x}\) and \(x=2 y .\) Find the plate's \begin{equation}\begin{array}{l}{\text { a. moment about the } x \text { -axis. }} \\ {\text { b. moment about the } y \text { -axis. }} \\ {\text { c. moment about the line } x=5 \text { . }} \\ {\text { d. moment about the line } x=-1} \\ {\text { e. moment about the line } y=2 \text { . }} \\\ {\text { f. moment about the line } y=-3 \text { . }} \\ {\text { g. mass. }} \\\ {\text { h. center of mass. }}\end{array}\end{equation}

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

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. $$y = ( 1 / 2 ) \left( x ^ { 2 } + 1 \right) , \quad 0 \leq x \leq 1 ; \quad y -axis$$

Find the center of mass of a thin plate covering the region between the \(x\) -axis and the curve \(y=2 / x^{2}, 1 \leq x \leq 2,\) if the plate's den- sity at the point \((x, y)\) is \(\delta(x)=x^{2}\)

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

The graph of \(y=x^{2}\) on \(0 \leq x \leq 2\) is revolved about the \(y\)-axis to form a tank that is then filled with salt water from the Dead Sea (weighing approximately 73 lb/ \(\mathrm{ft}^{3}\) . How much work does it take to pump all of the water to the top of the tank?

Find the center of mass of a thin plate covering the region bounded below by the parabola \(y=x^{2}\) and above by the line \(y=x\) if the plate's density at the point \((x, y)\) is \(\delta(x)=12 x\) .

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

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 \(\mathrm{lb} / \mathrm{ft}^{3}\). How much work is required to pump all of the fuel to a point 15 \(\mathrm{ft}\) above the top of the tank?

The region bounded by the curves \(y=\pm 4 / \sqrt{x}\) and the lines \(x=1\) and \(x=4\) is revolved about the \(y\) -axis to generate a solid. \begin{equation}\begin{array}{l}{\text { a. Find the volume of the solid. }} \\\ {\text { b. Find the center of mass of a thin plate covering the region if }} \\ {\text { the plate's density at the point }(x, y) \text { is } \delta(x)=1 / x \text { . }} \\ {\text { c. Sketch the plate and show the center of mass in your sketch. }}\end{array}\end{equation}

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

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. \(y=3 x, \quad y=0, \quad x=2\) a. The \(y\) -axis b. The line \(x=4\) c. The line \(x=-1\) d. The \(x\) -axis e. The line \(y=7\) f. The line \(y=-2\)

Emptying a water reservoir We model pumping from spherical containers the way we do from other containers, with the axis of integration along the vertical axis of the sphere. Use the figure here to find how much work it takes to empty a full hemispherical water reservoir of radius 5 \(\mathrm{m}\) by pumping the water to a height of 4 \(\mathrm{m}\) above the top of the reservoir. Water weighs 9800 \(\mathrm{N} / \mathrm{m}^{3} .\)

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

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.

The region between the curve \(y=2 / x\) and the \(x\) -axis from \(x=1\) to \(x=4\) is revolved about the \(x\) -axis to generate a solid. \begin{equation}\begin{array}{l}{\text { a. Find the volume of the solid. }} \\\ {\text { b. Find the center of mass of a thin plate covering the region if }} \\ {\text { the plate's density at the point }(x, y) \text { is } \delta(x)=\sqrt{x} \text { . }} \\ {\text { c. Sketch the plate and show the center of mass in your sketch. }}\end{array}\end{equation}

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

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. \(y=x^{3}, \quad y=8, \quad x=0\) $$ \begin{array}{ll}{\text { a. The } y \text { -axis }} & {\text { b. The line } x=3} \\ {\text { c. The line } x=-2} & {\text { d. The } x \text { -axis }} \\\ {\text { e. The line } y=8} & {\text { f. The line } y=-1}\end{array} $$

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 starlike 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 \(8 .\)

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

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. \(y=x+2, \quad y=x^{2}\) $$ \begin{array}{ll}{\text { a. The line } x=2} & {\text { b. The line } x=-1} \\\ {\text { c. The } x \text { -axis }} & {\text { d. The line } y=4}\end{array} $$

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 \begin{equation}\frac{d v}{d t}=\frac{d v}{d x} \frac{d x}{d t}=v \frac{d v}{d x}\end{equation} to show that the net work done by the force in moving the object from \(x_{1}\) to \(x_{2}\) is \begin{equation}W=\int_{x_{1}}^{x_{2}} F(x) d x=\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2},\end{equation} 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.

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

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. \(y=x^{4}, \quad y=4-3 x^{2}\) $$ \text { a. The }x=1 \quad \text { b. The x-axis} $$