## 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

- 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.
- Find the volume of the solid of revolution obtained by rotating the region bounded by \(y=x^2\), \(y=2x\) around the axis \(y=-2\).
- 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 \(y=3\).
- Find the volume of the solid of revolution obtained by rotating the triangle with vertices \((0,0)\), \((2,0)\), \((2,1)\) around the axis \(x=4\).
- 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.
- 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.)
- 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?
- 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 \(x=3\)?

### Textbook References

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