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AP Calculus AB: 10.6.5 The Washer Method across the y-Axis

Mathematics8 CardsCreated 8 months ago

This set covers finding volumes of solids of revolution with holes by rotating regions around the y-axis using the washer method. It stresses the importance of expressing radii and limits in terms of y, subtracting inner radius areas from outer radius areas, and integrating with respect to y (thickness dy) to compute volume accurately.

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The Washer Method across the y-Axis

Using the disk method, the volume V of a solid of revolution is given by , where R(y) is the radius of
the solid of revolution with respect to y.
• Washers are disks with smaller disks removed from the center.

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Key Terms

Term
Definition

The Washer Method across the y-Axis

Using the disk method, the volume V of a solid of revolution is given by , where R(y) is the radius of
the solid of revolution with respect to y...

note

  • Suppose you are given a solid of revolution defined by the four curves on the left. What is the volume of this solid of revolution?

What is the volume of the solid of revolution generated by revolving the area bounded by y=√x,y=0,and x=4 around the y-axis?

128π/5 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = x, y = −x + 2, and y = 0 about the y‑axis?

2π units^3

When the washer method is used to calculate the volume of a solid of revolution generated by revolving a planar region about the y‑axis, how thick is each washer?

dy

When the washer method is used to calculate the volume of a solid of revolution generated by rotating a planar region around the y‑axis, what is the variable of integration?

y

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AP Calculus AB: 10.6.5 The Washer Method across the y-Axis
TermDefinition

The Washer Method across the y-Axis

Using the disk method, the volume V of a solid of revolution is given by , where R(y) is the radius of
the solid of revolution with respect to y.
• Washers are disks with smaller disks removed from the center.

note

  • Suppose you are given a solid of revolution defined by the four curves on the left. What is the volume of this solid of revolution?

  • Start by graphing the four curves. The region described by the curves is a small piece that looks like a triangle someone sat on.

  • Rotating the region across the y-axis gives you the solid of revolution.

  • Slicing the solid vertically gives very strange shapes. But horizontal slices give washers. So you should try to find the volume using the washer method.

  • Remember, the washer method is just a modification of the disk method. Instead of integrating the entire area, you only integrate the area of the washer. To do that, you must subtract the area given by the inner radius from the area given by the outer radius.

  • Since the thickness of each disk is a little change in y, your integral needs to be expressed in terms of y. That means the limits of integration and the radius must be in terms of y instead of x.

  • It is a great idea to draw a small washer so you can better visualize which equations give which radii.

What is the volume of the solid of revolution generated by revolving the area bounded by y=√x,y=0,and x=4 around the y-axis?

128π/5 units^3

What is the volume of the solid of revolution generated by revolving the area bounded by y = x, y = −x + 2, and y = 0 about the y‑axis?

2π units^3

When the washer method is used to calculate the volume of a solid of revolution generated by revolving a planar region about the y‑axis, how thick is each washer?

dy

When the washer method is used to calculate the volume of a solid of revolution generated by rotating a planar region around the y‑axis, what is the variable of integration?

y

What is the volume of the solid of revolution generated by revolving the area bounded by y = −x^ 2 + 1, y = 0, and x = 0 about the y‑axis?

π/2 units^3

When computing the volume of a solid of revolution generated by revolving a region about the x‑axis, it is sometimes possible to use symmetry to simplify the integral by replacing the lower limit with 0 and doubling the result. When can this technique be used for revolutions about the y‑axis?

When the planar region is symmetric about the x‑axis.