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Section 3.4 Calculus 2

The Cylindrical Shell Method
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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