1.3: Volumes of Revolution - Cylindrical Shells
- Page ID
- 163251
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Deriving the Method of Cylindrical Shells
The volume of a single cylindrical shell is\[ V = 2 \pi r_i^* h_i \Delta r, \nonumber \]where \( r_i^* = \frac{1}{2}\left( r_{\text{Outer}} + r_{\text{Inner}} \right) \) and \( \Delta r = r_{\text{Outer}} - r_{\text{Inner}} \).
Proof
Let \(h(\lambda)\) be continuous and nonnegative. Define \(\mathbf{R}\) as the region bounded by the graph of \(h(\lambda)\), the \(\lambda\)-axis, the line \(\lambda=a\), and the line \(\lambda=b\). Then the volume of the solid of revolution formed by revolving \(\mathbf{R}\) around the \(\lambda\)-axis is given by\[V=\int_{\lambda = a}^{\lambda = b} 2 \pi \, r\left( \lambda \right) \, h\left(\lambda\right) \, d\lambda. \nonumber \]
Proof
Finding Volumes Using the Method of Cylindrical Shells
Compute the volume generated by rotating \(y=e^{x^2}\), \(0 \leq x \leq 1\) about the \(y\)-axis. Discuss the problems with trying to compute the volume generated by rotating this curve about the \(x\)-axis.
Setup, but do not compute, the integral to find the volume \(V\) generated by rotating the region bounded by the given curves about the \(y\)-axis.\[ y = 6\left( x-2 \right)^2, \quad y = x^2-4x+9 \nonumber \]
Find the volume \(V\) generated by rotating the region bounded by the given curves about the \(x\)-axis.\[ x=1+(y-2)^2, \quad x=2. \nonumber \]
Find the volume of the solid obtained by rotating the region bounded by\[ xy=1, \quad y=0, \quad x=1, \quad x=2\nonumber \]about \(x=-1\).
Setup, but do not compute, the integral to find the volume \(V\) generated by rotating the region bounded by the given curves about the line \(y=5\).\[x^2-y^2=7, \quad x=4\nonumber \]
Set up an integral for the volume of a solid torus with radii \(r\) and \(R\). By interpreting the integral as an area, find the volume of the torus.