Chapter 6

Find the center of mass of a thin plate of constant density \(\delta\) covering the given region. a. The region cut from the first quadrant by the circle \(x^{2}+y^{2}=9\) b. The region bounded by the \(x\) -axis and the semicircle \(y=\sqrt{9-x^{2}}\) Compare your answer in part (b) with the answer in part (a).

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

Find the volumes of the solids. The base of the solid is the disk \(x^{2}+y^{2} \leq 1 .\) The cross-sections by planes perpendicular to the \(y\) -axis between \(y=-1\) and \(y=1\) are isosceles right triangles with one leg in the disk.

Find the lengths of the curves. If you have a grapher, you may want to graph these curves to see what they look like. $$x=\int_{0}^{y} \sqrt{\sec ^{4} t-1} d t, \quad-\pi / 4 \leq y \leq \pi / 4$$

Find the center of mass of a thin plate of constant density \(\delta\) covering the given region. The "triangular" region in the first quadrant between the circle \(x^{2}+y^{2}=9\) and the lines \(x=3\) and \(y=3 .\) (Hint: Use geometry to find the area.)

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

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 \(\bar{x}=a>1 .\) Also, find \(\lim _{a \rightarrow \infty} \bar{x}\).

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(y\) -axis. $$y=3 /(2 \sqrt{x}), \quad y=0, \quad x=1, \quad x=4$$

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

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=x^{3} / 9, \quad 0 \leq x \leq 2 ; \quad x \text { -axis }$$

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. $$y=x^{2}, \quad-1 \leq x \leq 2$$

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 density at the point \((x, y)\) is \(\delta(x)=x^{2}\).

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.

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{x}, \quad 3 / 4 \leq x \leq 15 / 4 ; \quad x \text { -axis }$$

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. $$y=\tan x, \quad-\pi / 3 \leq x \leq 0$$

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 areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{2 x-x^{2}}, \quad 0.5 \leq x \leq 1.5 ; \quad x \text { -axis }$$

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. $$x=\sin y, \quad 0 \leq y \leq \pi$$

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. a. Find the volume of the solid. b. Find the center of mass of a thin plate covering the region if the plate's density at the point \((x, y)\) is \(\delta(x)=1 / x\). c. Sketch the plate and show the center of mass in your sketch.

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(x\) -axis. $$x=\sqrt{y}, \quad x=-y, \quad y=2$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=\sqrt{x+1}, \quad 1 \leq x \leq 5 ; \quad x \text { -axis }$$

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. $$x=\sqrt{1-y^{2}}, \quad-1 / 2 \leq y \leq 1 / 2$$

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(x\) -axis. $$x=y^{2}, \quad x=-y, \quad y=2, \quad y \geq 0$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$x=y^{3} / 3, \quad 0 \leq y \leq 1 ; \quad y \text { -axis }$$

A vertical right-circular cylindrical tank measures 9 \(\mathrm{m}\) high and \(6 \mathrm{m}\) in diameter. It is full of kerosene weighing \(7840 \mathrm{N} / \mathrm{m}^{3} .\) How much work does it take to pump the kerosene to the level of the top of the tank?

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. $$y^{2}+2 y=2 x+1 \quad \text { from } \quad(-1,-1) \text { to }(7,3)$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$x=(1 / 3) y^{3 / 2}-y^{1 / 2}, \quad 1 \leq y \leq 3 ; \quad y \text { -axis }$$

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. $$y=\sin x-x \cos x, \quad 0 \leq x \leq \pi$$

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(x\) -axis. $$x=2 y-y^{2}, \quad x=y$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$x=2 \sqrt{4-y}, \quad 0 \leq y \leq 15 / 4 ; \quad y-\text { axis }$$

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. $$y=\int_{0}^{x} \tan t d t, \quad 0 \leq x \leq \pi / 6$$

Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines about the \(x\) -axis. $$y=|x|, \quad y=1$$

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

A right-circular cylindrical tank of height 3 m and radius 1.5 m is lying horizontally and is full of diesel fuel weighing \(8300 \mathrm{N} / \mathrm{m}^{3}\). How much work is required to pump all of the fuel to a point \(4.5 \mathrm{m}\) 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. $$y=x^{3}, \quad y=0, \quad x=2$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. $$y=(1 / 2)\left(x^{2}+1\right), \quad 0 \leq x \leq 1 ; \quad y \text { -axis }$$

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

Find the areas of the surfaces generated by revolving the curves 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 \(\quad\) (Hint: \(\quad\) 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. $$y=x-x^{2}, \quad y=0$$

Find the areas of the surfaces generated by revolving the curves about the indicated axes. If you have a grapher, you may want to graph these curves to see what they look like. \(x=\left(y^{4} / 4\right)+1 /\left(8 y^{2}\right), \quad 1 \leq y \leq 2 ; \quad x\) -axis \(\quad\) (Hint: \(\quad\) Express \(d s=\sqrt{d x^{2}+d y^{2}}\) in terms of \(d y,\) and evaluate the integral \(S=\int 2 \pi y d s\) with appropriate limits.)

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 $$\frac{d v}{d t}=\frac{d v}{d x} \frac{d x}{d t}=v \frac{d v}{d x}$$ 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.

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

Find the moment about the \(x\) -axis of a wire of constant density that lies along the curve \(y=\sqrt{x}\) from \(x=0\) to \(x=2\).

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 \(y=-2\)

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

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 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 moment about the \(x\) -axis of a wire of constant density that lies along the curve \(y=x^{3}\) from \(x=0\) to \(x=1\).

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. \(y=x^{3}, \quad y=8, \quad x=0\) a. The \(y\) -axis b. The line \(x=3\) c. The line \(x=-2\) d. The \(x\) -axis e. The line \(y=8\) f. The line \(y=-1\)

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