Shell Method Calculator

Shell method calculator determining the surface area and volume of shells of revolution, when integrating along an axis perpendicular to revolution.

What Is Shell Method?

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

This method is useful whenever the washer method is very hard to carry out, generally, the representation of the inner and outer radii of the washer is difficult.

The volume of a cylinder of height h and radius r is πr^2 h.

How to Use The Shell Method?

The volume of the solid shell between two different cylinders, of the same height, one of radius and the other of radius.

The cylindrical shells calculator integrates of given function with step-wise calculation for the volume of solids.

Basic Shell Method Formulas:

The various shell method formulas depend on the axis of curves. 

• About Y-Axis

Spin around the area under the curve of f(x).

Volume = V = 2π ∫x f(x) dx

• About X-Axis

Spin around the area under the curve of f(y).

Volume = V = 2π ∫y f(y) dy

• Between Two Curves About the Y-Axis

Spinning around the area between two curves f(x) and g(x)

Volume = V = 2π ∫x[f(x) – g(x)] dx

• Between Two Curves About the X-Axis

Spinning around the area between two curves f(y) and g(y)

Volume = V = 2π ∫y[f(y) – g(y)] dy

• Between Two Curves About x = h

Spinning around the area between two curves f(x) and g(x)

Volume = V = 2π ∫(x – h) [f(x) – g(x)] dx

• Between Two Curves About y = k

Spinning around the area between two curves f(y) and g(y)

Volume = V = 2π ∫(y – k) [f(y) – g(y)] dy

Shell Method Example:

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

Solution:

Step 1:

Put integral In Shell Method Formula

$$\int (2 \pi x \left(3 x^{3} + 2 x^{2}\right))\, dx$$

The integral of a constant times a function is the constant times the integral of the function:

$$\int 2 \pi x \left(3 x^{3} + 2 x^{2}\right)\, dx = 2 \pi \int x \left(3 x^{3} + 2 x^{2}\right)\, dx$$

Step 2:

Rewrite The Integral:

$$x \left(3 x^{3} + 2 x^{2}\right) = 3 x^{4} + 2 x^{3}$$

Step 3:

Integrate Term-By-Term:

The integral of constant times a function is the constant times the integral of the function: $$\int 3 x^{4}\, dx = 3 \int x^{4}\, dx$$

Step 4:

Putting Limits By the Fundamental Theorem of Calculus

$$\int x^{4}\, dx = \frac{x^{5}}{5}$$

$$\int 2 x^{3}\, dx = 2 \int x^{3}\, dx$$

$$\int x^{3}\, dx = \frac{x^{4}}{4}$$

$$\frac{3 x^{5}}{5} + \frac{x^{4}}{2}$$

$$2 \pi \left(\frac{3 x^{5}}{5} + \frac{x^{4}}{2}\right)$$

$$\frac{\pi x^{4} \left(6 x + 5\right)}{5}$$

Step 6:

Simply Shell Method

$$\frac{\pi x^{4} \left(6 x + 5\right)}{5}+ \mathrm{constant}$$

=1591π5 ≈ 999.655

How Shell Method Calculator Works?

An online shell method volume calculator finds the volume of a cylindrical shell of revolution by following these steps:

Input:

• First, enter a given function.
• Now, substitute the upper and lower limits for integration.
• Hit the calculate button.

Output:

• The step-by-step solution.
• The standard shell method equation.

Reference:

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

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