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 viet.dang@humble.k12.tx.us 2/27/2019 7.3: Volume by Shells

Example of Volume by Shells Link 2/27/2019 7.3: Volume by Shells

Cylindrical Shells Method 2/27/2019 7.3: Volume by Shells

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) 2/27/2019 7.3: Volume by Shells

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: 2/27/2019 7:39 PM 12.2: Evaluating Limits

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 2/27/2019 7.3: Volume by Shells

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 2/27/2019 7.3: Volume by Shells

Example 2 Find the volume of the solid obtained by rotating about the y-axis the region between 𝒚=𝒙 and 𝒚= 𝒙 𝟐 . 2/27/2019 7.3: Volume by Shells

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. 2/27/2019 7.3: Volume by Shells

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. 2/27/2019 7.3: Volume by Shells

Assignment Page 462 1, 3, 13, 17, 21, 23 2/27/2019 7.3: Volume by Shells