# 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