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AP Calculus AB: 10.7.3 The Shell Method: Integrating with Respect to Y

Mathematics6 CardsCreated 3 months ago

This content explains how to use the shell method when integrating with respect to y, particularly for solids of revolution about the x-axis. It highlights choosing the shell method over washers when washers become complicated, detailing how to express radius, height, and thickness in terms of y and set up the integral accordingly.

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The Shell Method: Integrating with Respect to y

Using the shell method, the volume V of a solid of revolution is given by , where y is the radius and h(y)
is the height of an arbitrary shell.

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Key Terms

Term
Definition

The Shell Method: Integrating with Respect to y

Using the shell method, the volume V of a solid of revolution is given by , where y is the radius and h(y)
is the height of an arbitrary shell.<...

note

  • Consider the solid of revolution described on the left. To find the volume, it is a good idea to start by graphing the solid.

  • <...

Which of the following is the volume of the solid of revolution formed by revolving the region bounded by y=x2 and x=1 around the x-axis, where x≥0. Use the cylindrical shell method.

π/5

Consider a semicircle with radius R and center (0, 0). Which of the following is the volume of the solid of revolution generated by rotating the shaded area to the right of the y‑axis, bounded below by the chord y = c, and bounded above by the semicircle around the x‑axis in terms of the radius R of the circle only?

Use the cylindrical shell method. The line segment OB, which joins the center of the circle (0, 0) and the point of intersection of the chord y = c with the circle, makes a 30° angle with the x‑axis.

Note: The equation of a circle of radius R and center (0, 0) is x 2 + y 2 = R 2

√3/4πR^3

What is the volume of the solid of revolution formed by shifting the region bounded by the curve y = x 2, the line y = 0, and the line x = 1 over by one unit along the positive x‑axis and revolving the resulting region around the x‑axis?

Use the cylindrical shell method.

π/5

Evaluate the following as true or false. If the area bounded by the curve y=x2, the y-axis, and the line y=1 is rotated about the y-axis, then the volume of the solid of revolution is the same as that of the solid of revolution formed by rotation around the x-axis. Hint: Use the cylindrical shell method.

false

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AP Calculus AB: 10.7.3 The Shell Method: Integrating with Respect to Y
TermDefinition

The Shell Method: Integrating with Respect to y

Using the shell method, the volume V of a solid of revolution is given by , where y is the radius and h(y)
is the height of an arbitrary shell.

note

  • Consider the solid of revolution described on the left. To find the volume, it is a good idea to start by graphing the solid.

  • Notice that the region is revolved around the x-axis. Always make sure that you revolve the region about the correct line. Using the wrong axis of revolution will greatly change your graph.

  • Consider using the washer method on the solid of revolution first. From x = 0 to x = 1, the washer is defined by two lines. From x = 1 to x = 16, the washer is defined by a line and a parabola. The way the washer is defined changes. Washers might not be the easiest way to find the volume.

  • Now consider the shell method. Shells could be stacked around each other starting at y = 1, and running to y = 4. More importantly, the shells would always be defined by the y-axis and the parabola.

  • Looking at an arbitrary shell, you can see that the radius of the shell is equal to y. Use the radius to find the
    circumference of the shell.

  • Notice that the width of the shell is equal to x. The width must be expressed in terms of y since the thickness is a small change in y.

  • Multiply all the pieces together and you are ready to integrate.

  • Since the shells stack from 1 to 4, those are the limits of
    integration. Multiply the circumference, height, and thickness together to express the volume of an arbitrary shell.

  • Notice that the integration is pretty straightforward. Setting up the integral is the tricky part.

Which of the following is the volume of the solid of revolution formed by revolving the region bounded by y=x2 and x=1 around the x-axis, where x≥0. Use the cylindrical shell method.

π/5

Consider a semicircle with radius R and center (0, 0). Which of the following is the volume of the solid of revolution generated by rotating the shaded area to the right of the y‑axis, bounded below by the chord y = c, and bounded above by the semicircle around the x‑axis in terms of the radius R of the circle only?

Use the cylindrical shell method. The line segment OB, which joins the center of the circle (0, 0) and the point of intersection of the chord y = c with the circle, makes a 30° angle with the x‑axis.

Note: The equation of a circle of radius R and center (0, 0) is x 2 + y 2 = R 2

√3/4πR^3

What is the volume of the solid of revolution formed by shifting the region bounded by the curve y = x 2, the line y = 0, and the line x = 1 over by one unit along the positive x‑axis and revolving the resulting region around the x‑axis?

Use the cylindrical shell method.

π/5

Evaluate the following as true or false. If the area bounded by the curve y=x2, the y-axis, and the line y=1 is rotated about the y-axis, then the volume of the solid of revolution is the same as that of the solid of revolution formed by rotation around the x-axis. Hint: Use the cylindrical shell method.

false