Shell Method for Volumes of Solids of Revolution

Applications of Integration

Shell Method for Volumes of Solids of Revolution

The shell method is a powerful technique for finding the volume of a solid generated by revolving a region around a line (axis). It is especially useful when the region is bounded by functions that are easier to describe in terms of one variable, and when the axis of revolution is parallel to that variable.

• Shell Method Formula: The volume of a solid of revolution using the shell method is given by:

$V = \int_{a}^{b} 2\pi \cdot \text{(radius)} \cdot \text{(height)} \; dx$ $V = \int_{c}^{d} 2\pi \cdot \text{(radius)} \cdot \text{(height)} \; dy$

• Radius: The distance from the shell to the axis of revolution.

• Height: The length of the shell, determined by the difference between the bounding functions.

Example 1: Region Bounded by $y = x^2 + 1$, $y = 1$, and $x = 2$

• Region Description: The region R is bounded above by $y = x^2 + 1$, below by $y = 1$, and on the right by $x = 2$.

Revolving about the vertical line $x = 3$

• Radius: $3 - x$

• Height: $x^2 + 1 - 1 = x^2$

• Volume Integral:

$V = \int_{0}^{2} 2\pi (3 - x) (x^2) \, dx$

Revolving about the horizontal line $y = -1$

• Change of Variable: Solve $y = x^2 + 1 \Rightarrow x = \sqrt{y - 1}$

• Radius: $y - (-1) = y + 1$

• Height: $2 - \sqrt{y - 1}$

• Volume Integral:

$V = \int_{1}^{5} 2\pi (y + 1) (2 - \sqrt{y - 1}) \, dy$

Example 2: Triangular Region Bounded by $y = x$, $y = 12 - 2x$, and $y = \frac{3}{2} - x^2$

• Region Description: The region R is bounded by three lines/curves, forming a triangle.

Revolving about the line $x = 0$

• For $1 \leq x \leq 4$:

  • Radius: $x$

  • Height: $x - (\frac{3}{2} - x^2) = 3x^2 - \frac{3}{2}$

• For $4 \leq x \leq 7$:

  • Radius: $x$

  • Height: $(12 - 2x) - (\frac{3}{2} - x^2) = \frac{21}{2} - 3x^2$

• Volume Integral:

$V = \int_{1}^{4} 2\pi x (3x^2 - \frac{3}{2}) \, dx + \int_{4}^{7} 2\pi x (\frac{21}{2} - 3x^2) \, dx$

Revolving about the line $x = 8$

• For $1 \leq x \leq 4$:

  • Radius: $8 - x$

  • Height: $3x^2 - \frac{3}{2}$

• For $4 \leq x \leq 7$:

  • Radius: $8 - x$

  • Height: $\frac{21}{2} - 3x^2$

• Volume Integral:

$V = \int_{1}^{4} 2\pi (8 - x) (3x^2 - \frac{3}{2}) \, dx + \int_{4}^{7} 2\pi (8 - x) (\frac{21}{2} - 3x^2) \, dx$

Revolving about the line $y = -3$

• For $-2 \leq y \leq 1$:

  • Radius: $y + 3$

  • Height: $(6 - y^2) - (3 - 2y) = 3 + 3y^2$

• For $1 \leq y \leq 4$:

  • Radius: $y + 3$

  • Height: $(6 - y^2) - y = 6 - 3y^2$

• Volume Integral:

$V = \int_{-2}^{1} 2\pi (y + 3) (3 + 3y^2) \, dy + \int_{1}^{4} 2\pi (y + 3) (6 - 3y^2) \, dy$

Revolving about the line $y = 5$

• For $-2 \leq y \leq 1$:

  • Radius: $5 - y$

  • Height: $3 + 3y^2$

• For $1 \leq y \leq 4$:

  • Radius: $5 - y$

  • Height: $6 - 3y^2$

• Volume Integral:

$V = \int_{-2}^{1} 2\pi (5 - y) (3 + 3y^2) \, dy + \int_{1}^{4} 2\pi (5 - y) (6 - 3y^2) \, dy$

Summary Table: Shell Method Setup for Various Axes

Axis of Revolution	Variable	Radius	Height	Limits	Integral
x = 0	x	x	3x^2 - 3/2 (1 ≤ x ≤ 4), 21/2 - 3x^2 (4 ≤ x ≤ 7)	1 to 4, 4 to 7	$\int_{1}^{4} 2\pi x (3x^2 - \frac{3}{2}) \, dx + \int_{4}^{7} 2\pi x (\frac{21}{2} - 3x^2) \, dx$
x = 8	x	8 - x	3x^2 - 3/2 (1 ≤ x ≤ 4), 21/2 - 3x^2 (4 ≤ x ≤ 7)	1 to 4, 4 to 7	$\int_{1}^{4} 2\pi (8 - x) (3x^2 - \frac{3}{2}) \, dx + \int_{4}^{7} 2\pi (8 - x) (\frac{21}{2} - 3x^2) \, dx$
y = -3	y	y + 3	3 + 3y^2 (-2 ≤ y ≤ 1), 6 - 3y^2 (1 ≤ y ≤ 4)	-2 to 1, 1 to 4	$\int_{-2}^{1} 2\pi (y + 3) (3 + 3y^2) \, dy + \int_{1}^{4} 2\pi (y + 3) (6 - 3y^2) \, dy$
y = 5	y	5 - y	3 + 3y^2 (-2 ≤ y ≤ 1), 6 - 3y^2 (1 ≤ y ≤ 4)	-2 to 1, 1 to 4	$\int_{-2}^{1} 2\pi (5 - y) (3 + 3y^2) \, dy + \int_{1}^{4} 2\pi (5 - y) (6 - 3y^2) \, dy$

Key Points and Applications

• Shell method is often preferred when the axis of revolution is not a boundary of the region, or when integrating with respect to the other variable would be more complicated.

• Always identify the radius and height for each shell, and set up the limits of integration based on the region's boundaries.

• Do not evaluate the integral unless specifically required; setting up the correct integral is often the main task.

Example Application: Calculating the volume of a vase-shaped solid by revolving a region bounded by curves about a vertical or horizontal axis.

Additional info: The shell method is part of the broader topic of applications of integration, specifically for finding volumes of solids of revolution. It complements the disk/washer method and is useful for more complex regions or axes.