Volumes of Revolution

YOU WILL LEARN ABOUT:

How to use the washer method to find the volume of a shape that is obtained by rotating an appropriate plane region about the x- or y-axis.

The formula for the volume of the solid of revolution that has washers as its cross section is given by

$π \int_{a}^{b} (r_{o} {(x)}^{2} - r_{i} {(x)}^{2}) d x$ , if the axis of rotation is the x-axis.
$π \int_{a}^{b} (r_{o} {(y)}^{2} - r_{i} {(y)}^{2}) d y$ , if the axis of rotation is the y-axis.

Note that r_o gives the radius of the outer region of the washer and r_i gives the radius of the inner region. Also note that r_o (x) and r_i (x) are the radii of the washers rotated about the x-axis and that r_o (y) and r_i (y) are the radii of the washers rotated about the y-axis.

*The washer method is an application of the method of slicing.

VOLUMES OF REVOLUTION - WASHER METHOD

To rotate grid, click and drag

REVOLVE REGION

# OF SLICES

AXIS OF REVOLUTION

x y

Region 1

Region 2

Region 3

$Region 1 \begin{array}{l} y_{1} = \{\begin{cases} 0.5, \\ x - 0.5, \end{cases} & \begin{matrix} 0 \leq x \leq 1 \\ 1 < x < 3 \end{matrix} \\ y_{2} = \{\begin{cases} 1, \\ x, \end{cases} & \begin{matrix} 0 \leq x \leq 1 \\ 1 < x \leq 3 \end{matrix} \end{array}$

$Region 1 \begin{array}{l} x_{1} = \{\begin{cases} 0, \\ y, \end{cases} & \begin{array}{l} 0.5 \leq y < 1.0 \\ 1.0 \leq y \leq 3.0 \end{array} \\ x_{2} = \{\begin{cases} y + 0.5, \\ 3, \end{cases} & \begin{matrix} 0.5 \leq y \leq 2.5 \\ 2.5 < y \leq 3.0 \end{matrix} \end{array}$

$Volume \int_{a}^{b} π (y_{2}^{2} - y_{1}^{2}) d x \approx$

$Volume \int_{a}^{b} π (x_{2}^{2} - x_{1}^{2}) d x \approx$

$0$

$Region 2 \begin{array}{l} y = \sqrt{1 - {(x - 1.5)}^{2}}, \\ 0.5 \leq x \leq 2.5 \end{array}$

$Region 2 \begin{array}{l} x_{1} = 1.5 - \sqrt{1 - y^{2}}, - 1 \leq y \leq 1 \\ x_{2} = 1.5 + \sqrt{1 - y^{2}}, - 1 \leq y \leq 1 \end{array}$

$Volume \int_{a}^{b} π y^{2} d x \approx$

$Volume \int_{a}^{b} π (x_{2}^{2} - x_{1}^{2}) d x \approx$

$0$

$Region 3 \begin{array}{l} y_{1} = \sqrt{x}, & 0 \leq x \leq 1 \\ y_{2} = x^{2}, & 0 \leq x \leq 1 \end{array}$

$Region 3 \begin{array}{l} x_{1} = y^{2}, & 0 \leq y \leq 1 \\ x_{2} = \sqrt{y}, & 0 \leq y \leq 1 \end{array}$

$Volume \int_{a}^{b} π (y_{2}^{2} - y_{1}^{2}) d x \approx$

$Volume \int_{a}^{b} π (x_{2}^{2} - x_{1}^{2}) d x \approx$

$0$

$\begin{array}{ll} Number of Slices : & 0 \\ Width of Slices : & 0.00 \\ Sum of Volume of Slices \approx & 0.00 units3 \end{array}$

YOU WILL LEARN ABOUT:

Click on the tabs on the right to watch videos on rotating a region about the x-axis and y-axis.

Click the Revolve Region button and watch the animation before using the slider bars.

Click the Revolve Region button and watch the animation before using the slider bars.

Click the Revolve Region button and watch the animation before using the slider bars.