AI - Volume by Disks and Washers Lesson

Volume by Disks and Washers

Returning to the loaf of bread analogy, each bread slice has a slightly different area, which results in different slices having slightly different volumes. For a solid that is not a right circular cylinder, the volume can be estimated by slicing the solid into pieces and approximating each piece by a cylinder and applying equation image indicator $LaTeX: V=\pi Ah$ $V=\pi Ah$ . The technique for computing volume of a solid using cross sections perpendicular to an axis of revolution is called the disk method.

Let the radius of each bread slice be f(x) and each height (thickness) be equation image indicator $LaTeX: \Delta x$ $\Delta x$ . Imagine generating your own bread by revolving a rectangle of dimensions f(x) and $LaTeX: \Delta x$ $\Delta x$ about the x-axis as x varies along [a, b]. The resulting cross section closely resembles a right circular cylinder but all of the pieces are not necessarily congruent. This means that different slices have slightly different volumes; however, the loaf's total volume is still the sum of the volumes of all of the slices.

Did you know? Around 200 B.C. Archimedes (the one who famously jumped out of his bath and ran down the street shouting "Eureka! I've got it!") found volumes of spheres by slicing and adding the volumes of each slice using V=πr²h Based on the volume formula for a right circular cylinder $LaTeX: \left(V=\pi r^2h\right)$ $\left(V=\pi r^2h\right)$ , each bread slice has volume equation image indicator $LaTeX: V=\pi f\left(x_i\right)^2\Lambda x$ $V=\pi f\left(x_i\right)^2\Lambda x$ . The total volume is then approximated by summing the volumes of each cylinder as the number of cylinders grows towards infinity.

$LaTeX: V=\lim_{x \to \infty}\sum_{i=1}^{n} \pi\left[f\left(x_i\right)\right]^2\Delta x$ $V=\lim_{x \to \infty}\sum_{i=1}^{n} \pi\left[f\left(x_i\right)\right]^2\Delta x$ . With a < x< b, this limiting sum is written as equation image indicator $LaTeX: V=\int_a^b\pi\left[f\left(x\right)\right]^2dx\:or\:V=\int_a^b\pi\left(radius\right)^2dx\:or\:V=\int_a^bArea\left(x\right)dx$ $V=\int_a^b\pi\left[f\left(x\right)\right]^2dx\:or\:V=\int_a^b\pi\left(radius\right)^2dx\:or\:V=\int_a^bArea\left(x\right)dx$ .

View the series of three presentations on solids created by revolving a region about the x-axis.

The disk method may also be used to determine the volume of solids generated by revolving a planar region about the y-axis. As before, a cross section perpendicular to the axis of revolution is needed. Note that when the disk method is used and a sketch of a typical strip (slice) is drawn on the graph of the region, the typical strip is always perpendicular to the axis of revolution and its height (thickness) always corresponds to the variable of the axis of revolution. In general, the volume of a solid of revolution obtained by revolving a planar region about a line is calculated using two basic defining formulas: equation image indicator $LaTeX: V=\int_a^bArea\left(x\right)dx\:or\:V=\int_a^bArea\left(y\right)dy$ $V=\int_a^bArea\left(x\right)dx\:or\:V=\int_a^bArea\left(y\right)dy$ .

View the two videos illustrating revolution about the y-axis.

The presentation below shows how the disk method is applied in varied situations to find volumes of solids of revolution.

Washer Method

image of washer method graphed When the region to be revolved about a line does not border on or cross the axis of revolution, the resulting solid has the shape of a washer with an outer radius and an inner radius instead of a disk. The technique for computing the volume of a region not bordering on the axis of revolution that uses cross sections perpendicular to the axis of revolution is called the washer method.

The presentation below develops an intuitive formula for finding the volume of a solid of revolution with a hole in it and illustrates how the washer method is applied to revolution about the x-axis and about a line other than one of the coordinate axes.

View the following presentations featuring the washer method for solids revolved about the x-axis and the y-axis.

Volume by Disks and Washers Practice

1. Determine the volume of the solid obtained by rotating the region bounded by equation image indicator LaTeX: y=x^2-2x $y=x^2-2x$ and $y=x$ about the line LaTeX: y=4 $y=4$ using the washer method. text annotation indicator

Solution: $LaTeX: \frac{153\pi}{5}$ $\frac{153\pi}{5}$

2. Determine the volume of the solid obtained by rotating the portion of the region bounded by equation image indicator $LaTeX: y=\sqrt[3]{x}$ $y=\sqrt[3]{x}$ and $LaTeX: y=\frac{x}{4}$ $y=\frac{x}{4}$ that lies in the first quadrant about the y-axis using the disk method.

Solution: $LaTeX: \frac{512\pi}{12}$ $\frac{512\pi}{12}$ text annotation indicator

Volume by Disks and Washers: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the volumes by disks and washers.

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