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AP Calculus AB: 10.6.2 The Disk Method along the y-Axis

Mathematics9 CardsCreated 3 months ago

This set focuses on using the disk method to find the volume of solids of revolution formed by rotating a region around the y-axis. It highlights the importance of integrating with respect to y, converting functions accordingly, and setting limits based on y-values to calculate the volume through integration of circular cross-sectional areas.

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The Disk Method along the y-Axis

Revolving a plane region about a line forms a solid of revolution.
• 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.

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Key Terms

Term
Definition

The Disk Method along the y-Axis

Revolving a plane region about a line forms a solid of revolution.
• Using the disk method, the volume V of a solid of revolution is given by , ...

note

  • In this example you are asked to find the volume of a
    solid of revolution rotated around the y-axis instead of the x-axis.

  • <...

What is the volume of the solid generated by revolving the plane region bounded by y = x 2, y = 1, and y = 4 in the first quadrant about the y-axis?

15π/2

Which of these integrals defines the volume of the solid that is generated by revolving the plane region bounded by y = x 3 and x = 0 about the y‑axis from y = 1 to y = 8?

∫^8_1 π(3√y)^2dy

Set up the integral that produces the volume of the solid generated by revolving the plane region bounded by y = x 5, y = 1, y = 5, and x = 0 about the y-axis.

∫^5_1 πy^2/5dy

Find the volume of the solid generated by revolving the plane region bounded by
y = x 3, y = 1, y = 8, and x = 0 about the y-axis.

93π/5

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AP Calculus AB: 10.6.2 The Disk Method along the y-Axis
TermDefinition

The Disk Method along the y-Axis

Revolving a plane region about a line forms a solid of revolution.
• 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

  • In this example you are asked to find the volume of a
    solid of revolution rotated around the y-axis instead of the x-axis.

  • Notice that instead of slicing vertically like you would with a rotation about the x-axis, this solid seems easier to solve using the disk method with respect to the y-axis.

  • Notice that the thickness of an arbitrary slice of the solid is given by a small change in y, instead of a small change in x.

  • That’s a clue that you should integrate with respect to y for this problem.

  • Using the disk method to find the volume of vertically stacked disks is much like using the method for horizontally stacked disks.

  • To integrate with respect to y you must express all of your functions in terms of y instead of x.

  • The disks span from y = 1 to y = 8, so these are the limits of integration. Notice that the limits of integration are also in terms of y.

  • The volume of each disk is equal to the area of the disk times the thickness. Remember, the area of a circle is given by the formula A = r 2 .

  • Integrate this product, and the result is the volume of the solid of revolution.

What is the volume of the solid generated by revolving the plane region bounded by y = x 2, y = 1, and y = 4 in the first quadrant about the y-axis?

15π/2

Which of these integrals defines the volume of the solid that is generated by revolving the plane region bounded by y = x 3 and x = 0 about the y‑axis from y = 1 to y = 8?

∫^8_1 π(3√y)^2dy

Set up the integral that produces the volume of the solid generated by revolving the plane region bounded by y = x 5, y = 1, y = 5, and x = 0 about the y-axis.

∫^5_1 πy^2/5dy

Find the volume of the solid generated by revolving the plane region bounded by
y = x 3, y = 1, y = 8, and x = 0 about the y-axis.

93π/5

Find the volume of the solid generated by revolving the plane region bounded by
y = x/2, y = 1, y = 2, and x = 0 about the y-axis.

28π/3

Given a region that is bounded by y = 1/x 3, y = 1, y = 2, and x = 0, set up (but do not evaluate) the integral for the volume of the solid generated by revolving the region around the y-axis.

∫^2_1 πy^−2/3dy

A region is shown below. Set up the integral for determining the volume of the solid generated by revolving the region around the y-axis.

∫^b_a π{[f(y)]^2−[g(y)]^2}dy