Study Notes on Volumes by Cylindrical Shells

Applications of Integration

Introduction to Volumes by Cylindrical Shells

  • Focuses on the method of finding volumes of solids obtained by rotating a region around an axis, specifically the y-axis.

  • Involves situations where rotating around the y-axis is more manageable than around the x-axis.

The Method of Cylindrical Shells

Definition and Description

  • Cylindrical Shell: A hollow cylinder formed by rotating a vertical rectangle around the y-axis.

  • Components:

    • Inner Radius ($r_1$)

    • Outer Radius ($r_2$)

    • Height ($h$)

![Figure 2](Figure 2)

Volume Calculation

  • The volume ($V$) of a cylindrical shell can be computed by subtracting the volume of the inner cylinder ($V1$) from the volume of the outer cylinder ($V2$):
    V = V2 - V1

Formula Derivation

  • Define $Δr = r2 - r1$ (thickness of the shell) and the average radius $r_{avg}$.

  • The formula for the volume can also be expressed using circumference:

    • V = [ ext{circumference}] imes [ ext{height}] imes [ ext{thickness}]

Integration Setup for Volumes

Solid S Description

  • Define solid $S$ obtained by rotating the region bounded by the functions:

    • Lower boundary: $y = 0$

    • Upper boundary: $y = f(x)$, where $f(x) ext{ is non-negative}$

    • Left boundary: $x = a$

    • Right boundary: $x = b$ ($b > a ext{ and } a ext{ are non-negative}$)

![Figure 3](Figure 3)

Gridding the Interval

  • Divide the interval $[a, b]$ into $n$ subintervals: $[x{i-1}, xi]$.

  • Set $x_i$ to be at the midpoint of the $i^{th}$ interval.

Shell Volume Computation

  • Rotating a rectangle with base $[x{i-1}, xi]$ and height $f(x_i)$ produces a cylindrical shell:

    • Average radius = $x_i$

    • Height = $f(x_i)$

    • Thickness = $Δx$

![Figure 4](Figure 4)

Approximation of Total Volume

  • The total volume $V$ of $S$ is approximated by summing the volumes of the shells:

    • V ≈ ext{sum}igto{} V = 2 ext{π} ext{(average radius)} imes ext{(height)} imes ext{(thickness)}

  • As $n o ∞$, this approximation approaches the exact volume of solid $S$.

Conclusion of Volume Formula

  • The final volume obtained by this method for solid $S$ is given by:

    • V = ext{Volume of the solid}= ext{integral}igg(2 ext{π}x f(x)igg)dx from $a$ to $b$.

  • This formula is often memorized by visualizing a typical shell cut and flattened (illustrated in Figure 5).

![Figure 5](Figure 5)

Examples

Example 1

  • Problem Statement: Find the volume of the solid obtained by rotating a bounded region around the y-axis.

  • Solution:

    • A typical shell characteristics:

    • Radius = $x$

    • Circumference = $2πx$

    • Height = $f(x)$

![Figure 6](Figure 6)

Validating the Shell Method

  • It can be verified that utilizing the shell method yields the same result as the washer method.

Disks and Washers versus Cylindrical Shells

Choosing the Correct Method

  • Considerations when deciding between disks/washers or cylindrical shells:

    • Easier boundary descriptions: top and bottom functions $y=f(x)$ vs. left and right boundaries $x=g(y)$.

    • The complexity of limits of integration for either variable.

    • Whether one method requires two separate integrals while the other does not.

    • The ability to evaluate the integral based on variable choice.

Visual Representation

  • Draw a sample rectangle to visualize the region:

    • Thickness corresponds to $Δx$ (for vertical rectangles) or $Δy$ (for horizontal rectangles).

    • Revolving leads to either disks/washers or cylindrical shells depending on orientation.

Example 5

Problem Statement

  • The region is in the first quadrant bounded by the curves (specific functions not provided) and $y = 2x$, rotating about $x = -1$.

Solution Approach
(a) Using x as the Variable of Integration

  • Illustrate the sample rectangle vertically:

    • As this will generate cylindrical shells.

(b) Using y as the Variable of Integration

  • Illustrate the sample rectangle horizontally:

    • This approach leads to washer-shaped cross-sections.