Shell Method Calculator

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by Alan Walker

Last Updated February 07, 2023

Introduction to Shell Method Calculator

When integrating along an axis perpendicular to the axis of revolution, an cylindrical shell method calculator determines the surface area and volume of revolution shells. This cylindrical shells calculator integrates a given function and calculates the volume of solids in a step-by-step manner. Learn how to use integration to find the area and volume of shells.

The cylindrical shell method is a mathematical methodology for estimating the volumes of revolution. This method comes in handy when the washer method is difficult to implement, such as representing the washer's inner and outer radii is difficult.

Learn what is partial fraction in integration and how important are online calculators for learning integration online.

How Does the Shell Method Calculator Work?

Following these procedures, an online shell method volume calculator calculates the volume of a cylindrical shell of revolution:


  • First, select a function
  • Now, for integration, use the upper and lower limits
  • Click Calculate

These are the simple inputs of cylindrical shell method calculator. User needs to add them carefully and once its done, the method of cylindrical shells calculator provides an accurate output in form of results.


  • The cylindrical shells calculator provides a step-by-step methodology for determining the volume using definite and indefinite integration.
  • Using the conventional shell technique equation, this calculator does precise shell calculations.

Related: Find useful online calculators like iterated integral calculator and triple integral calculator with steps for your learning and practice online.

Another thing to keep in mind before finding volume in this shell method solver, we may need to judge by self which method is good fit on our problem. So we have to take decision when to use washer or shell method according to the situation.

When calculating capacity is the length of the area which will be taken into account. The height of the region will be utilized if you want to get the surface. Using a shell method volume calculator, only solids of revolution are examined, which produces a volume output.

Aside from that, this technique calculates volume using a three-dimensional axis. When an object revolves a cross-section in the XY-plane around the y-axis using this method, it specifies the cylinder shape as it moves vertically.

Furthermore, imagine that the area is cylinder-shaped. In that scenario, the volume will be determined by multiplying the cross-sectional area by the height, as well as the inner and outer radius of the cylinder. The volume by cylindrical shells calculator uses shell method formula to determine the size of a solid:


Related: Find step by step integral calculator on home page and also find more about integration and its importance.

Formulas used by Shell Method Calculator with steps

The cylindrical shells volume calculator uses two different formulas. It uses shell volume formula (to find volume) and another formula to get the surface area. Both formulas are listed below:

  1. shell volume formula
  2. $$V=(R2-r2)*L*PI$$

    Where R=outer radius, r=inner radius and L=length

  3. Shell surface area formula
  4. $$A=2*PI*(R+r)*(R-r+L)$$

    Where r= inner radius, R=outer radius and L= length


Solve 3x3+2x2 by shell method with upper limit 2 and lower limit


$$ -\frac{1591π}{5} $$

Indefinite Integral

$$ \frac{πx^4(6x+5)}{5} \;+\;constant $$

Also find more useful tools on this website like definite integration calculator and indefinite integral calculator with steps for your practice regarding indefinite and definite integration concepts.


Visualize a vertical cut of a specific region and then open it to make a flat plate as another method to think about the shape with a narrow vertical slice. After using this cylindrical shell method calculator to determine the volume of a solid, you can use the graphical depiction to visualize your problem.

Provide the inner and outer radius, as well as the length and height, to plot the graph. When you use this calculator, you will get the graphical representation of the cylindrical shape.

From the foregoing explanation, you have a clear understanding of how cylindrical shells formula works for various solid shapes. And how to use shell method formula calculator to find the volume of the shell.

Related: Also find other useful online tools like washer integral calculator and disk washer method calculator on this website.

Frequently Asked Questions

What's the distinction between the disc technique and the Shell method?

While the shell approach involves building rings of varied shapes and radii that are defined by the revolution r(x) along the x coordinate at each x, the disc method involves stacking discs of varying shapes and radii that are defined by the revolution r(x) along the x coordinate at each x.

This website provides handful for integration tools which you can use to increase your productivity just like inverse laplace transform calculator & inverse fourier transform calculator online.

When should I use the washer or the shell method?

x(y)d is integrated if y is given as a washer. y(x)dx is integrated, and Y is expressed as a shell, Which is easier depending on the function assigned to it.

How to solve equation 3x3+2x2 by shall method with upper limit 2 and lower limit 3.

We can solve this equation as:

$$ -\frac{4169π}{5} $$

Indefinite Integral

$$ \frac{πx^5(5x+4)}{5}\;+\;constant $$

Graphical representation:

Shell method graphical representation

What is the definition of a mathematical spherical shell?

A spherical shell is a geometric generalization of a three-dimensional ring. The region of the ball between two concentric circles of different radii is called this.

What is cylindrical shell method?

The shell method is an integration method to find the volume of a solid of resolution. It integrates a function perpendicular to the axis of resolution and finds the volume by decomposing the solid into cylindrical shells. The shell method formula is,

$$ V \;=\; 2 \pi \int_a^b r(x)h(x) dx $$


  • r(x)represents distance from the axis of rotation to x.
  • h(x)represents the height of the shell.

The cylindrical shell calculator allow you to calculate shell volume online for free.

When to use cylindrical shell method?

Although the disk method is more efficient than the shell method. But both cannot help when you are finding volume of revolution of complex functions. It is because the disk method is used when the curve is revolved around the x-axis but the shell method is used when the curve is revolved around y-axis. And you can solve this using cylindrical shell volume calculator free online.

What is the volume of a shell?

The volume of a shell formed by the revolution about an axis of rotation. It depends on the radius, height and thickness of cylindrical shell. The shell integration is used to find the volume of a shell. The formula of shell volume is,

$$ V \;=\; 2 \pi \int_a^b r(x)h(x) dx $$

With this volume of revolution shell method calculator, you can calculate any equation easily without an error.

What is the area of the shell method?

In shell method, the best way to determine the area of the shell is to cut and lay it out flat that form a rectangle. So, it means that the area of the shell is the area of rectangle formed by the shell that is



  • r= is the radius of the shell.
  • h= is the height of the shell.

Use this volume by shells calculator for your equation and their step by step solution.

  • Extremely useful. It makes it very easy to calculate via shell method
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