6.6E: Exercises for Section 6.6
- 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}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
- 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.