Synthesis: Choose Your Method

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We practice choosing a method for computing volume when none is specified.

Online Texts

Ximera: Comparing Washers and Shells

Examples

(Spring 2016 Final Exam) Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y = 1-x^2$ , $y = 0$ , and $x = 0$ about the line $x=2$ .

Using $x$ as the slicing variable, slices are to the axis $x=2$ . This alignment corresponds to the disk or washershell method. (Expand to continue.)

The height of a shell is
$2$ $2-x$ $0$ $x-2$ $1-x^2$ $1 - x^2 - 2$
and the radius is
$2$ $2-x$ $0$ $x-2$ $1-x^2$ $1 - x^2 - 2$

Therefore $V = \int _{\answer {0}}^{\answer {1}} \answer {2 \pi (2-x)(1-x^2)} dx = \answer {\frac {13 \pi }{6}}.$

(Fall 2012 Final Exam) The region in the $xy$ -plane bounded by $y = (x-1)^{1/4}$ and the $x$ -axis for $1 \leq x \leq 2$ is rotated about the $x$ -axis. What is the volume of the resulting solid of revolution?

Using $x$ as the slicing variable, slices are to the axis $x=2$ . This alignment corresponds to the disk or washershell method. (Expand to continue.)

The outer radius of a disk is
$2$ $y^4+1$ $0$ $(x-1)^{1/4}$
and the inner radius is
$2$ $y^4+1$ $0$ $(x-1)^{1/4}$

Therefore $V = \int _{\answer {1}}^{\answer {2}} \answer {\pi (x-1)^{1/2}} dx = \answer {\frac {2 \pi }{3}}.$

Find the volume of the solid generated by revolving the region in the first quadrant bounded on the left by $y = \sqrt {x}$ , on the right by $y = x-2$ , and below by $y = 0$ about the axis $y = 0$ .

To carry out the calculation as efficiently as possible, the $x$ $y$ -variable should be used as the variable for slicing. (Expand to continue.)

Using $y$ as the slicing variable, slices are to the axis $y=0$ . This alignment corresponds to the disk or washershell method. (Expand to continue.)

The height of a shell is
$y + 2$ $y^2$ $x-2$ $\sqrt {x}$ $y^2-y-2$ $y+2-y^2$
and the radius is
$y$ $y^2$ $x-2$ $\sqrt {x}$ $y^2-y-2$ $y+2-y^2$

Therefore $V = \int _{\answer {0}}^{\answer {2}} \answer {2 \pi y(y+2-y^2)} dy = \answer {\frac {16 \pi }{3}}.$