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Section 3.3 Calculus 2

The Washer Method
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3.3 The Washer Method

3.3.1 Rotation about Horizontal Axes

  • Many solids may be described as revolutions of two-dimensional regions. Such solids have washer-shaped cross-sections.
  • To obtain the volume of a solid of revolution about a horizontal axis, identify the outer radius \(R(x)\) and inner radius \(r(x)\) for each \(x\)-value, the leftmost \(a\) and rightmost \(b\) \(x\)-values in the region and use the formula \(V=\int_a^b \pi([R(x)]^2-[r(x)]^2)\,dx\).
  • Example Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,0)\), \((2,2)\), \((2,4)\) around the \(x\)-axis.
  • Example Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=x\) and \(y=x^2\) around the axis \(y=2\).

3.3.2 Rotation about Vertical Axes

  • When the axis of revolution is vertical, simply use functions of \(y\) rather than \(x\), and the bottommost \(c\) and topmost \(d\) \(y\)-values: \(V=\int_c^d \pi([R(y)]^2-[r(y)]^2)\,dy\).
  • 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 axis \(x=-1\).

Review Exercises

  1. Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((3,0)\), \((3,3)\), \((0,3)\) around the \(x\)-axis.
  2. Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=x^2\), \(y=2x\) around the axis \(y=-2\).
  3. Consider the region in the \(xy\) plane satisfying \(|x|\leq\frac{\pi}{2}\) and \(|y|\leq\cos x\). Find the volume of the solid of revolution obtained by rotating this region around the axis \(y=3\).
  4. Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,0)\), \((2,0)\), \((2,1)\) around the axis \(x=4\).
  5. Find the volume of the solid of revolution obtained by rotating the region bounded by \(x+y=1\), \(y=\ln x\), \(y=1\) around the \(y\)-axis.
  6. Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((-\sqrt 2,0)\), \((0,\sqrt 2)\), \((\sqrt 2,0)\) around the axis \(y=\sqrt 2-x\). (Hint: Translate the region and its axis so that it has a horizontal or vertical axis of revolution.)
  7. What integral is produced by the washer method for the volume of the solid of revolution obtained by rotating the region bounded by \(y=x^2\) and \(y=4\) around the \(x\)-axis?
  8. What integral is produced by the washer method for the volume of the solid of revolution obtained by rotating the triangle with vertices \((1,1)\), \((2,1)\), \((2,0)\) around the axis \(x=3\)?

Solutions 1-3

Solutions 4-6

Solutions 7-8

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 6.1