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Table of Content
Shell method calculator determining the surface area and volume of shells of revolution, when integrating along an axis perpendicular to revolution.
“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.
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.
The various shell method formulas depend on the axis of curves.
Spin around the area under the curve of f(x).
Volume = V = 2π ∫x f(x) dx
Spin around the area under the curve of f(y).
Volume = V = 2π ∫y f(y) dy
Spinning around the area between two curves f(x) and g(x)
Volume = V = 2π ∫x[f(x) – g(x)] dx
Spinning around the area between two curves f(y) and g(y)
Volume = V = 2π ∫y[f(y) – g(y)] dy
Spinning around the area between two curves f(x) and g(x)
Volume = V = 2π ∫(x – h) [f(x) – g(x)] dx
Spinning around the area between two curves f(y) and g(y)
Volume = V = 2π ∫(y – k) [f(y) – g(y)] dy
Calculate the shell method about the y-axis if f(x) = 2x^2+3x^3 and the interval is {2, 3}.
$$\int (2 \pi x \left(3 x^{3} + 2 x^{2}\right))\, dx$$
The integral of a constant times a function is the constant times the integral of the function:
$$\int 2 \pi x \left(3 x^{3} + 2 x^{2}\right)\, dx = 2 \pi \int x \left(3 x^{3} + 2 x^{2}\right)\, dx$$
$$x \left(3 x^{3} + 2 x^{2}\right) = 3 x^{4} + 2 x^{3}$$
The integral of constant times a function is the constant times the integral of the function:
$$\int 3 x^{4}\, dx = 3 \int x^{4}\, dx$$
$$\int x^{4}\, dx = \frac{x^{5}}{5}$$
$$\int 2 x^{3}\, dx = 2 \int x^{3}\, dx$$
$$\int x^{3}\, dx = \frac{x^{4}}{4}$$
$$\frac{3 x^{5}}{5} + \frac{x^{4}}{2}$$
$$2 \pi \left(\frac{3 x^{5}}{5} + \frac{x^{4}}{2}\right)$$
$$\frac{\pi x^{4} \left(6 x + 5\right)}{5}$$
$$\frac{\pi x^{4} \left(6 x + 5\right)}{5}+ \mathrm{constant}$$
=1591π5 ≈ 999.655
An online shell method volume calculator finds the volume of a cylindrical shell of revolution by following these steps:
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.