Shell Method Calculator

Shell Method Calculator:

Use this shell method calculator to find the volume of rotating solids using cylindrical shells, with detailed integration steps to make the process easy to follow and understand.

What Is the Shell Method?

“In mathematics, the technique of calculating the volumes of revolution is called the cylindrical shell method.”

The shell method is very useful when the axis of revolution is parallel to the axis of integration. In this case, the washer method is difficult to apply because of complex expressions for the inner and outer radii.

Shell Method Image

The Shell Method Formula:

The general formulas for the volume using the shell method are:

V=∫_a^b 2πr(x)h(x)dx (for revolution about the y-axis or a vertical line)

V=∫_c^d 2πr(y)h(y)dy (for revolution about the x-axis or a horizontal line)

The shell method has different formulas to find volume of a solid of revolution. These formulas depend on the axis of rotation and the function(s) that define the region.

1. Rotation About the Y-Axis:

Consider the region bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b (where a ≤ b). When this region is revolved about the y-axis, the volume of the resulting solid is given by:

V=2π∫_a^b x⋅f(x)dx

Where,

x is the radius of a cylindrical shell
f(x) is the height of the cylinder

2. Rotation About the X-Axis:

Consider the region bounded by the curve x = f(y), the y-axis, and the horizontal lines y = c and y = d (where c ≤ d). Whenever this region rotates about the x-axis, the volume of solid is determined by:

V=2π∫_c^d y⋅f(y)dy

Where,

y is the radius of a cylindrical shell
f(y) is its height

3. About the Y-Axis (Between Two Curves):

Imagine you have a region bounded by the curves y = f(x), y = g(x) (where f(x) ≥ g(x)), and the vertical lines x = a and x = b. When revolved about the y-axis, the volume is found by:

V=2π∫_a^b x⋅[f(x) - g(x)]dx

Where,

x represents the radius
[f(x) - g(x)] is the height of a cylindrical shell

4. About the X-Axis (Between Two Curves):

A region bounded by the curves x = f(y), x = g(y) (where f(y) ≥ g(y)), and the horizontal lines y = c and y = d. When it revolves around the x-axis, the volume is:

V = 2π∫_c^d y⋅[f(y)–g(y)]dy

Where,

y shows the radius
[f(y)–g(y)] indicates the height of a cylindrical shell

5. Rotation About a Vertical Line x = h:

Imagine a region bounded by y = f(x), y = g(x), and vertical lines x = a and x = b. When it revolved around x = h, the radius of a shell is ∣x-h∣. The volume is:

V = 2π∫_a^b ∣x - h∣⋅[f(x) - g(x)]dx

6. Rotation About a Horizontal Line y =k:

Consider the region bounded by x = f(y), x = g(y), and horizontal lines y = c and y = d. When revolved about y = k, the radius of a shell is ∣y-k∣. The volume is:

V = 2π∫_c^d ∣y–k∣⋅[f(y)–g(y)]dy

For quick and easy calculations of the volume of a solid of revolution, use our volume shell method calculator. It incorporates the formulas mentioned above to find the volume.

Want to see how shell method formulas are applied? Check out the following example:

Shell Method Example:

Calculate the shell method about the y-axis if f(x) = 3x³ + 2x² and the interval is {2, 3}.

Solution:

⟹Step 1: Put the integral into the Shell Method formula
= ∫(2πx(3x³ + 2x²)) dx

The integral of a constant times a function is the constant times the integral of the function:
= ∫2πx(3x³ + 2x²) dx = 2π ∫x(3x³ + 2x²) dx

⟹Step 2: Rewrite the integral
= x(3x³ + 2x²) = 3x⁴ + 2x³

⟹Step 3: Integrate term by term
= ∫3x⁴ dx = 3 ∫x⁴ dx
= ∫2x³ dx = 2 ∫x³ dx

⟹Step 4: Apply the power rule
= ∫x⁴ dx = (x⁵) / 5
= ∫x³ dx = (x⁴) / 4

So we get:
= 3(x⁵ / 5) + 2(x⁴ / 4) = (3x⁵ / 5) + (x⁴ / 2)

Multiply by 2π:
= 2π((3x⁵ / 5) + (x⁴ / 2))

⟹Step 5: Simplify
= (πx⁴(6x + 5)) / 5

⟹Step 6: Final Shell Method result (plus constant of integration)
= (πx⁴(6x + 5)) / 5 + C

If you evaluate it at a certain x:
= (1591π) / 5 ≈ 999.655

How To Use the Shell Method Calculator?

Here are the steps to use our cylindrical shell method calculator accurately:

Enter the function(s) defining the boundary of the region
Substitute the upper and lower limits for integration
Select the variable of integration (W.R.T. - With Respect To) from the drop-down menu
Click on the “Calculate” button, and the calculator will then find the volume of the solid of revolution using the shell method, providing you with the detailed steps

FAQ’s:

When To Use The Shell Method?

When you see that the axis of rotation is parallel to the axis of integration
- Rotating around a vertical line (e.g., y-axis, x = a), ⟹ integrates with respect to x
- Rotating around a horizontal line (e.g., x-axis, y = b), ⟹ integrate with respect to y
When you find out that solving the bounding function is difficult or impossible
If it seems like the integral that you will get from the shell method will be in a simpler form

What’s The Difference Between The Shell Method And The Disk/Washer Method?

These two methods are used to find the volume of a solid of revolution, but they use two separate integration processes to slice the region.

Shell Method: In this method, a solid is cut into cylindrical shells (parallel to the axis of rotation) to find the volume
Disk/Washer Method: In this method, the solid is cut into slices (perpendicular to the axis of rotation). When the slices are solid, then the disk equation (V_d) is used, and when the slice contains a hole or space, then the washer equation (V_w) is used

Can I Use The Shell Method Around Any Axis?

Yes, it is possible to use the shell method around any axis (horizontal, vertical, or even other lines). The important step is to correctly represent the radius and height of the cylindrical shells.

References:

From the source of Wikipedia: shell integration, integral calculus, disc integration, and the axis of revolution.

From the source of Paul's Notes: Volume With Cylinders, method of cylinders, method of shells, method of rings/disks.