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AP Calculus AB: 10.7.2 Why Shells Can Be Better Than Washers

Mathematics6 CardsCreated 3 months ago

This set explains scenarios where the shell method simplifies volume calculations compared to the washer method, especially when washers require splitting integrals or complicated algebra. It emphasizes how shells avoid multiple integrals by stacking cylindrical layers and demonstrates setting up integrals using radius, height, and thickness in terms of x.

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Why Shells Can Be Better Than Washers

•Using the shell method, the volume V of a solid of revolution is given by
, where x is the radius and
h(x) is the height of an arbitrary shell.
• For some solids the shell method of finding volume is simpler than the washer method.

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Key Terms

Term
Definition

Why Shells Can Be Better Than Washers

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

note

  • Consider the solid of revolution described on the left. Start by graphing it.

  • Notice that if you use the washer method to ev...

Find the volume of the solid of revolution generated by rotating about y = a the region bounded by the loop of the given relation 2ay 2 = x (a − x)2 for 0 ≤ x ≤ a and a > 0

8√2/15πa^3

Consider a region bounded by curves y = x and y = x 2 rotated about the x‑axis. What is the volume of the resulting solid?

2π/15

What is the volume of the solid of revolution obtained by rotating the region bounded by the curves x = y 3 − y 4 and x = 0 about the line y = −2?

4π/15

True or false?
Consider the area bounded by the functions y = f (x), or x = g ( y), and the lines x = a and x = b, for b > a ≥ 0 and d > c ≥ 0 as shown. The volume of the solid of revolution generated by sweeping the region around the y‑axis can best be calculated using the washer method. The thickness of the washers should be dy, and you should use the product of the cross-sectional area of the washers and the elemental thickness dy.

false

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AP Calculus AB: 10.7.2 Why Shells Can Be Better Than Washers
TermDefinition

Why Shells Can Be Better Than Washers

•Using the shell method, the volume V of a solid of revolution is given by
, where x is the radius and
h(x) is the height of an arbitrary shell.
• For some solids the shell method of finding volume is simpler than the washer method.

note

  • Consider the solid of revolution described on the left. Start by graphing it.

  • Notice that if you use the washer method to evaluate the volume, the washers would stack vertically. The integral would need to be expressed in terms of y, which requires you to do some algebra.

  • There is another reason not to use the washer method. Notice that the washers near the bottom of the solid are defined differently than the washers at the top. If you used washers to find the volume, you would have to evaluate two integrals.

  • Try using the shell method instead.

  • To use the shell method, start by drawing an arbitrary shell.

  • Remember that shells are simply curled up rectangles. By uncurling the shell you can find a simple way of describing its volume.

  • Notice that the radius of the shell is x. Use the radius to find the circumference of the shell. The circumference of the shell is equal to the base of the rectangular solid.

  • The height of the shell is equal to the height of the rectangle. In this example the height can be found by using the equation of the curve.

  • The thickness of the shell is a small change in x. The shells range from x = 1 to x = 2.

  • Now you have all the pieces you need to find the volume. The actual integral is the easy step here.

  • You avoid having to break the integral into two pieces by using the shell method.

Find the volume of the solid of revolution generated by rotating about y = a the region bounded by the loop of the given relation 2ay 2 = x (a − x)2 for 0 ≤ x ≤ a and a > 0

8√2/15πa^3

Consider a region bounded by curves y = x and y = x 2 rotated about the x‑axis. What is the volume of the resulting solid?

2π/15

What is the volume of the solid of revolution obtained by rotating the region bounded by the curves x = y 3 − y 4 and x = 0 about the line y = −2?

4π/15

True or false?
Consider the area bounded by the functions y = f (x), or x = g ( y), and the lines x = a and x = b, for b > a ≥ 0 and d > c ≥ 0 as shown. The volume of the solid of revolution generated by sweeping the region around the y‑axis can best be calculated using the washer method. The thickness of the washers should be dy, and you should use the product of the cross-sectional area of the washers and the elemental thickness dy.

false