6.6E: Exercises for Section 6.6

Last updated
Save as PDF

Page ID: 70167

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

In exercises 1 - 6, calculate the center of mass for the collection of masses given.

1) $m_1=2$ at $x_1=1$ and $m_2=4$ at $x_2=2$

2) $m_1=1$ at $x_1=−1$ and $m_2=3$ at $x_2=2$

Answer: $x = \frac{5}{4}$

3) $m=3$ at $x=0,1,2,6$

4) Unit masses at $(x,y)=(1,0),(0,1),(1,1)$

Answer: $\left(\frac{2}{3},\, \frac{2}{3}\right)$

5) $m_1=1$ at $(1,0)$ and $m_2=4$ at $(0,1)$

6) $m_1=1$ at $(1,0)$ and $m_2=3$ at $(2,2)$

Answer: $\left(\frac{7}{4},\,\frac{3}{2}\right)$

In exercises 7 - 16, compute the center of mass $\bar x.$

7) $ρ=1$ for $x∈(−1,3)$

8) $ρ=x^2$ for $x∈(0,L)$

Answer: $\dfrac{3L}{4}$

9) $ρ=1$ for $x∈(0,1)$ and $ρ=2$ for $x∈(1,2)$

10) $ρ=\sin x$ for $x∈(0,π)$

Answer: $\frac{π}{2}$

11) $ρ=\cos x$ for $x∈\left(0,\frac{π}{2}\right)$

12) $ρ=e^x$ for $x∈(0,2)$

Answer: $\dfrac{e^2+1}{e^2−1}$

13) $ρ=x^3+xe^{−x}$ for $x∈(0,1)$

14) $ρ=x\sin x$ for $x∈(0,π)$

Answer: $\dfrac{π^2−4}{π}$

15) $ρ=\sqrt{x}$ for $x∈(1,4)$

16) $ρ=\ln x$ for $x∈(1,e)$

Answer: $\frac{1}{4}(1+e^2)$

In exercises 17 - 19, compute the center of mass $(\bar{x},\bar{y}).$ Use symmetry to help locate the center of mass whenever possible.

17) $ρ=7$ in the square $0≤x≤1, \; 0≤y≤1$

18) $ρ=3$ in the triangle with vertices $(0,0), \, (a,0)$, and $(0,b)$

Answer: $\left(\frac{a}{3},\, \frac{b}{3}\right)$

19) $ρ=2$ for the region bounded by $y=\cos(x), \; y=−\cos(x), \; x=−\frac{π}{2}$, and $x=\frac{π}{2}$

In exercises 20 - 26, use a calculator to draw the region, then compute the center of mass $(\bar{x},\bar{y}).$ Use symmetry to help locate the center of mass whenever possible.

20) [T] The region bounded by $y=\cos(2x), \; x=−\frac{π}{4}$, and $x=\frac{π}{4}$

Answer: $\left(0,\frac{π}{8}\right)$

21) [T] The region between $y=2x^2, \; y=0, \; x=0,$ and $x=1$

22) [T] The region between $y=\frac{5}{4}x^2$ and $y=5$

Answer: $(0,3)$

23) [T] Region between $y=\sqrt{x}, \; y=\ln x, \; x=1,$ and $x=4$

24) [T] The region bounded by $y=0$ and $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$

Answer: $\left(0,\frac{4}{π}\right)$

25) [T] The region bounded by $y=0, \; x=0,$ and $\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$

26) [T] The region bounded by $y=x^2$ and $y=x^4$ in the first quadrant

Answer: $\left(\frac{5}{8},\, \frac{1}{3}\right)$

In exercises 27 - 31, use the theorem of Pappus to determine the volume of the shape.

27) Rotating $y=mx$ around the $x$-axis between $x=0$ and $x=1$

28) Rotating $y=mx$ around the $y$-axis between $x=0$ and $x=1$

Answer: $V = \frac{mπ}{3}$ units³

29) A general cone created by rotating a triangle with vertices $(0,0), \, (a,0),$ and $(0,b)$ around the $y$-axis. Does your answer agree with the volume of a cone?

30) A general cylinder created by rotating a rectangle with vertices $(0,0), \, (a,0), \, (0,b),$ and $(a,b)$ around the $y$-axis. Does your answer agree with the volume of a cylinder?

Answer: $V = πa^2b$ units³

31) A sphere created by rotating a semicircle with radius $a$ around the $y$-axis. Does your answer agree with the volume of a sphere?

In exercises 32 - 36, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $(\bar{x},\bar{y})$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

32) [T] Quarter-circle: $y=\sqrt{1−x^2}, \; y=0$, and $x=0$

Answer: $\left(\frac{4}{3π},\, \frac{4}{3π}\right)$

33) [T] Triangle: $y=x, \; y=2−x$, and $y=0$

34) [T] Lens: $y=x^2$ and $y=x$

Answer: $\left(\frac{1}{2},\, \frac{2}{5}\right)$

35) [T] Ring: $y^2+x^2=1$ and $y^2+x^2=4$

36) [T] Half-ring: $y^2+x^2=1, \; y^2+x^2=4,$ and $y=0$

Answer: $\left(0,\, \frac{28}{9π}\right)$

37) Find the generalized center of mass in the sliver between $y=x^a$ and $y=x^b$ with $a>b$. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.

38) Find the generalized center of mass between $y=a^2−x^2, \; x=0$, and $y=0$. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.

Answer: Center of mass: $\left(\frac{a}{6},\,\frac{4a^2}{5}\right),$
Volume: $\dfrac{2πa^4}{9}$ units³

39) Find the generalized center of mass between $y=b\sin(ax),\; x=0,$ and $x=\dfrac{π}{a}.$ Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.

40) Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $a$ is positioned with the left end of the circle at $x=b, \, b>0,$ and is rotated around the $y$-axis.

$This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.$

Answer: Volume: $V = 2\pi^2a^2(b+a)$

41) Find the center of mass $(\bar{x},\bar{y})$ for a thin wire along the semicircle $y=\sqrt{1−x^2}$ with unit mass. (Hint: Use the theorem of Pappus.)

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.