Last Updated on October 10, 2025 by Rajeev Bagra

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When a 2-D region is revolved around an axis, it creates a 3-D solid. To find the volume of such a solid, we slice it into many thin pieces, find the volume of each slice, and then integrate (add up) all slices.

There are two main techniques for this in definite integral calculus: the Disk/Washer Method and the Shell Method.

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🧠 Intuitive Understanding

Disk (or Washer) Method

• Slices are made perpendicular to the axis of revolution.
• Each slice forms a disk (or washer if there’s a hole).
• The volume is the sum (integral) of these disk volumes.

Think of stacking thin coins or CDs along an axis.

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Shell Method

• Slices are made parallel to the axis of revolution.
• Each slice forms a thin cylindrical shell when revolved.
• The total volume is obtained by adding up all these shell volumes.

Think of many paper tubes (cylindrical shells) wrapped around the axis.

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🔢 Mathematical Formulas

Disk Method (No Hole)

For a region bounded by ( y = f(x) ), revolved about the x-axis:

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Washer Method (With Inner and Outer Radii)

For a region between ( y = R(x) ) and ( y = r(x) ) revolved about the x-axis:

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Shell Method (Around the y-axis)

For a region under ( y = f(x) ) from ( x=a ) to ( x=b ):

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Shell Method (Around the x-axis)

If the region is described by ( x = f(y) ) from ( y=c ) to ( y=d ):

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📊 Conceptual Diagrams

Disk / Washer Method

   y
   ↑
   |      *****
   |    *       *      y = f(x)
   |   *         *
   |    *       *
   |      *****
   +------------------> x
       |  |   |   |
       Thin slices (disks) perpendicular to x-axis

Here’s the same text rewritten with proper LaTeX shortcodes so it renders correctly:

Each slice has a radius and area .
Volume of slice ≈ .

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Shell Method

   y
   ↑
   |        ______
   |       |      |   ← vertical strip at x; when revolved, forms a thin shell
   |       |      |
   +------------------> x
           x   x+dx

When this strip at (x) (height (f(x))) revolves around the y-axis:

• Circumference = (2.pi.x)
• Height = (f(x))
• Thickness = (dx)

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🧮 Worked Examples

Example 1 — Disk Method

Find the volume formed by revolving the region under ( y = x^2 ) from ( x = 0 ) to ( x = 1 ) about the x-axis.

The volume using the disk method is:

Evaluating the integral:

Hence, the volume of the solid formed by revolving ( y = x^2 ) about the x-axis is π/5 cubic units.

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Example 2 — Shell Method

Revolve the same region ( y = x^2 ), ( 0 \le x \le 1 ), around the y-axis.

The volume using the shell method is:

Evaluating the integral:

Hence, the volume of the solid formed by revolving ( y = x^2 ) about the y-axis is π/2 cubic units.

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Example 3 — Choosing Between Disk and Shell

For the region bounded by , , and , revolved about the x-axis:

• Disk method (simple):

Shell method (harder):
Would require expressing ( x ) as a function of ( y ) and integrating with respect to ( y ).

Hence, Disk/Washer method is the better choice here.

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🧩 When to Use Which

Situation	Use Disk/Washer	Use Shell
Axis of rotation is x-axis and function is (y=f(x))	✅	❌
Axis of rotation is y-axis and function is (y=f(x))	❌	✅
Easy to express radius	✅
Easy to express height × radius		✅
Cross-sections perpendicular to axis	✅
Cross-sections parallel to axis		✅

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📘 Summary of Key Formulas

Method	Formula	Axis
Disk		x-axis
Washer		x-axis
Shell		y-axis
Shell		x-axis

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🪄 Final Thoughts

• Disk/Washer → use when perpendicular slicing gives easy radii.
• Shell → use when parallel slicing gives easier height × radius.
• If one looks messy, try the other — both represent the same geometric idea: summing infinitely thin volumes to find total volume.

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