# Section 3.3 Calculus 2

The Washer Method

## 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