1. Section 7.3 Calculus AP/Dual, Revised ©2016 viet.dang@humble.k12.tx.us
Volume by Shells
Section 7.3
Calculus AP/Dual, Revised ©2016
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7.3: Volume by Shells

2. Example of Volume by Shells
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7.3: Volume by Shells

3. Cylindrical Shells Method
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7.3: Volume by Shells

4. Cylindrical Shells Method
Its volume 𝑽 is calculated by subtracting the volume 𝑽𝟏 of the inner cylinder from the volume of the outer cylinder 𝑽𝟐.
Let ∆𝒓 = 𝒓𝟐 – 𝒓𝟏 (thickness of the shell) and (average radius of the shell).
Radius: 𝒓= (𝒓 𝟏 + 𝒓 𝟐 ) 𝟐
Volume: 𝒂 𝒃 𝟐𝒙 𝒇 𝒙 𝒅𝒙 or as known as Volume = (Circumference) (Height) (Thickness)
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7.3: Volume by Shells

5. Formula Here’s the best way to remember the formula.
Think of a typical shell, cut and flattened, with radius 𝒙, circumference 𝟐𝝅𝒙, height 𝒇 𝒙 , and thickness ∆𝒙 or 𝒅𝒙:
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12.2: Evaluating Limits

6. Example 1
Find the volume of the solid obtained by rotating about the 𝒚-axis the region bounded by 𝒚=𝟐 𝒙 𝟐 − 𝒙 𝟑 and 𝒚=𝟎
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7.3: Volume by Shells

7. Example 1
Find the volume of the solid obtained by rotating about the 𝒚-axis the region bounded by 𝒚=𝟐 𝒙 𝟐 − 𝒙 𝟑 and 𝒚=𝟎
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7.3: Volume by Shells

8. Example 2
Find the volume of the solid obtained by rotating about the y-axis the region between 𝒚=𝒙 and 𝒚= 𝒙 𝟐 .
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7.3: Volume by Shells

9. Example 3
Use cylindrical shells to find the volume of the solid obtained by rotating the region about the 𝒙-axis the region under the curve 𝒚= 𝒙 from 0 to 1.
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7.3: Volume by Shells

10. Example 4
Find the volume of the solid obtained by rotating the region bounded by 𝒚=𝒙− 𝒙 𝟐 and 𝒚=𝟎 about the line 𝒙=𝟐.
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7.3: Volume by Shells

11. Assignment
Page 462 1, 3, 13, 17, 21, 23
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7.3: Volume by Shells