2.R: Chapter 2 Review Exercises
- Page ID
- 196022
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)True or False? Justify your answer with a proof or a counterexample.
1) The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.
- Answer
- False
2) If the force is constant, the amount of work to move an object from \(x=a\) to \(x=b\) is \(F(b−a)\).
For exercises 3 - 6, use the requested method to determine the volume of the solid.
3) The volume that has a base of the ellipse \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\) and cross-sections of an equilateral triangle perpendicular to the \(y\)-axis. Use the method of slicing.
- Answer
- \(V = 32\sqrt{3}\, \text{units}^3\)
4) \(y=x^2−x\), from \(x=1\) to \(x=4\), rotated around the \(y\)-axis using the washer method
5) \(x=y^2\) and \(x=3y\) rotated around the \(y\)-axis using the washer method
- Answer
- \(V = \frac{162π}{5}\, \text{units}^3\)
6) \(x=2y^2−y^3,\; x=0\),and \(y=0\) rotated around the \(x\)-axis using cylindrical shells
For exercises 7 - 12, find
a. the area of the region,
b.the volume of the solid when rotated around the \(x\)-axis, and
c. the volume of the solid when rotated around the \(y\)-axis. Use whichever method seems most appropriate to you.
7) \(y=x^3,x=0,y=0\), and \(x=2\)
- Answer
- a. \(A = 4\) units2
b. \(V = \frac{128π}{7}\) units3
c. \(V = \frac{64π}{5}\) units3
8) \(y=x^2−x\) and \(x=0\)
9) [T] \(y=\ln(x)+2\) and \(y=x\)
- Answer
- a. \(A \approx 1.949\) units2
b. \(V \approx 21.952\) units3
c. \(V = \approx 17.099\) units3
10) \(y=x^2\) and \(y=\sqrt{x}\)
11) \(y=5+x, y=x^2, x=0\), and \(x=1\)
- Answer
- a. \(A = \frac{31}{6}\) units2
b. \(V = \frac{452π}{15}\) units3
c. \(V = \frac{31π}{6}\) units3
12) Below \(x^2+y^2=1\) and above \(y=1−x\)
For exercises 13 - 14, find the requested arc lengths. Use technology to approximate the length if the integral is impossible to evaluate.
13) The length of \(f(x)=\cos(x)\) from \(x=0\) to \(x=2\).
14) The length of \(g(y)=3−\sqrt{y}\) from \(y=0\) to \(y=4\)
- Answer
- \(s = \big[\sqrt{17}+\frac{1}{8}\ln(33+8\sqrt{17})\big]\) units
For exercises 15 - 16, find the surface area and volume when the given curves are revolved around the specified axis.
15) The shape created by revolving the region between \(y=4+x, \;y=3−x, \;x=0,\) and \(x=2\) rotated around the \(y\)-axis.
16) The loudspeaker created by revolving \(y=\dfrac{1}{x}\) from \(x=1\) to \(x=4\) around the \(x\)-axis.
- Answer
- Volume: \(V = \frac{3π}{4}\) units3
Surface area: \(A = π\left(\sqrt{2}−\sinh^{−1}(1)+\sinh^{−1}(16)−\frac{\sqrt{257}}{16}\right)\) units2