Shell Method Calculator

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

This cylindrical shells calculator does integration of given function with step-wise calculation for the volume of solids.

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 Shell Method?

The volume of the solid shell between two different cylinders, of the same height, one of radius and the other of radius r^2 > r^1 is π(r_2^2 –r_1^2) h = 2π r_2 + r_1 / 2 (r_2 – r_1) h = 2 πr △rh, where, r = ½ (r_1 + r_2) is the radius and △r = r_2 – r_1 is the change in radius.

If a profile b = f(a), for (a) between x and y is rotated about the y quadrant, then the volume can be approximated by the Riemann sum method of cylinders:

Every cylinder at the position x* is the width △a and height b = f(a*): so every component of the Riemann sum has the form 2π x* f(x*) △a.
In the limit when the value of cylinders goes to infinity, the Riemann sum becomes an integral representation of the volume V:

$$ V = ∫_a^b 2 π x y (dx) = V = ∫_a^b 2 π x f (x) dx $$

If the area between two different curves b = f(a) and b = g(a) > f(a) is revolved around the y-axis, for x from the point a to b, then the volume is:

$$ ∫_a^b 2 π x (g (x) – f (x)) dx $$

Now, this tool computes the volume of the shell by rotating the bounded area by the x coordinate, where the line x = 2 and the curve y = x^3 about the y coordinate.

Here y = x^3 and the limits are x = [0, 2].
The integral is:

$$ ∫_0^2 2 π x y dx = ∫_0^2 2 π x (x^3)dx $$

$$ = 2π∫_0^2x^4 = 2π [ x^5 / 5]_0^2 = 2π 32/5 = 64/5 π$$
The curves meet at the point x = 0 and at the point x = 1, so the volume is:

$$ ∫_0^1 2 π x (\sqrt{x} – x^2 )dx $$

$$= 2 π ∫_0^1(x^{3/2} – x^3)dx$$

$$= 2 π[ 2/5 x^{5/2} – x^4 / 4]_0^1$$
$$= 2 π (2 / 5 – 1 / 4) = 3 / 10 π$$

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 limit for integration.
• Hit the calculate button.

Output:

• The shell method calculator displays the definite and indefinite integration for finding the volume with a step-by-step solution.
• This calculator does shell calculations precisely with the help of 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.

From the source of Ximera: Slice, Approximate, Integrate, expand the integrand, parallel to the axis.