\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 3.4 Calculus 2


The Cylindrical Shell Method

3.4 The Cylindrical Shell Method

3.4.1 Rotation about Vertical Axes

  • As an alternative to the washer method, one may consider “cylindrical shell” cross-sections instead.
  • The lateral surface area of a cylinder is given by \(2\pi rh\).
  • For a vertical axis of revolution, identify the radius \(r(x)\) and height \(h(x)\) of a cylindrical shell, identify the leftmost \(a\) and rightmost \(b\) \(x\)-values of the region, and use the formula \(V=\int_a^b 2\pi r(x)h(x)\,dx\).
  • Example Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=0\), \(x=1\), \(y=\sqrt{x}\) around the line \(x=-1\).
  • Example Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=0\), \(x=\pm1\), \(y=x^2+1\) around the line \(x=2\).

3.4.2 Rotation about Horizontal Axes

  • When the axis of revolution is horiztonal, simply use functions of \(y\) rather than \(x\), and the bottommost \(c\) and topmost \(d\) \(y\)-values: \(V=\int_c^d 2\pi r(y)h(y)\,dy\).
  • Example Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((-1,2)\), \((0,1)\), \((2,2)\) around the \(x\)-axis.

Review Exercises

  1. Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,2)\), \((1,0)\), \((1,2)\) around the axis \(x=2\).
  2. Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=4\), \(y=x^2-4x+4\) around the \(y\)-axis.
  3. Find the volume of the solid of revolution obtained by rotating the region bounded by \(x=y^2-1\), \(x=3\) around the axis \(x=-1\).
  4. Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((4,2)\), \((2,6)\), \((0,6)\) around the axis \(y=2\).
  5. Find the volume of the solid of revolution obtained by rotating the region bounded by \(x=e\), \(y=2\), \(y=\ln x\) around the \(x\)-axis.
  6. Use the cylindrical shell method to prove the volume formula for a sphere: \(V=\frac{4}{3}\pi R^3\).
  7. What integral is produced by the cylindrical shell method for the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,0),(2,0),(0,4)\) around the \(y\)-axis?
  8. What integral is produced by the cylindrical shell method for the volume of the solid of revolution obtained by rotating the region bounded by \(x=0,y=2,x=y^3\) around the axis \(y=-1\)?

Solutions 1-3

Solutions 4-8


Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 6.2