Shell Method Calculator



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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 below mentioned 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 together as it uses one formula to find voluume and another formula to get the surface area. Both formulas are listed below:

  1. Formula for finding volume
  2. $$V=(R2-r2)*L*PI$$

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

  3. Formula for getting surface area
  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 the cylinder formula works for various solid shapes and how to use this calculator to calculate the volume of the shell.

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


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

Answer: 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?

Answer: 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.

Solve equation 3x3+2x2 by shall method with upper limit 2 and lower limit 3.


$$ -\frac{4169π}{5} $$

Indefinite Integral

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

Graphical representation:

Shell method graphical representation

What is the definition of a mathematical spherical shell?

Answer: 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.

Alan Walker

Alan Walker

Last Updated November 30, 2021

I am Mathematician, Tech geek and a content writer. I love solving patterns of different math queries and write in a way that anyone can understand. Math and Technology has done its part and now its the time for us to get benefits from it.