## Introduction to Cylindrical Shell Method Calculator:

Shell method calculator is an online tool that helps you evaluate the volume of solids of revolution in two-dimensional space. Our tool **finds the function** that rotates around a region's vertical or horizontal axis in a cross-section area.

It is a useful online tool that makes the process of calculating the volume of solids simpler as it takes a lot of time to calculate manually. Use our cylindrical shells calculator to cope with such problems before they arise.

## What is the Shell Method?

Shell method is a process that is used to calculate the **volume of a solid** of revolution in cylindrical volume. It has a function f(x) that rotates along the x-axis and y-axis in a bounded region over 2-D space.

Shell method uses the integration method to find the area under a bounded region. The axis of rotation depends on the given function's behavior or nature.

## Formula Used for the Shell Method:

The shell method has different types of functions for different **volume rotations** in a bounded region along the x-axis or y-axis.

**Area under the curve along the x-axis:**: If the given function is continuous over a bounded region [a, b]. Then the volume V which is rotating around the area under the curve about the y-axis

$$ V \;=\; \lim_{\Delta x \to 0} \sum_{i=0}^{n-1} 2 \pi x_i f(x_i) \Delta x \;=\; \int_a^b 2 \pi x f(x) dx $$

**Area between Curves— Integration w.r.t. X**: If f and g are continuous on the interval [a, b] for all x in [a, b] and the area bounded above f and below g and the lines x = a and x = b, then the volume V about the y-axis is:

$$ V \;=\; \lim_{\Delta x \to 0} \sum_{i = 0}^{n-1} 2 \pi x_i [f(x_i) - g(x_i)] \Delta x \;=\; \int_a^b 2\pi x [f(x) - g(x)] dx $$

**Area between Curves — Integration w.r.t. Y**: If ๐ and ๐ are non-negative and continuous on the interval [๐, ๐] for all ๐ฆ. The area bounded right by ๐ and left by g and the lines y = c and y = d then the volume V about the ๐ฅ-axis is:

$$ V \;=\; \lim_{\Delta y \to 0} \sum_{i=0}^{n-1} 2\pi y_i [f(y_i) - g(y_i)] \Delta y \;=\; \int_c^d 2\pi y [f(y) - g(y)] dy $$

## How to Find Volume Using the Shell Method?

Shell method is used to **find the volumes** of solids of revolution by integrating the volumes of cylindrical shells along the axis of rotation which may be around the x-axis or y-axis. Let's see the shell method working procedure of finding volume in steps.

**Step 1:** Identify the given function, the interval over which it is rotating, and the axis of rotation along the x-axis or y-axis.

**Step 2:** According to the axis of rotation, choose the appropriate formula for finding volume in the shell method. If the function is rotating along the y-axis.

$$ V \;=\; \int_a^b 2 \pi x [f(x) - g(x)] dx $$

If the function is rotating along the x-axis,

$$ V \;=\; \int_a^b 2\pi y [f(y) - g(y)] dy $$

**Step 3:** Adjust the given function and get the required values that are needed for your required formula of rotation.

**Step 4:** Put the given function value in the formula and integrate the function. For integration, you can use any method to get a solution.

**Step 5:** Apply the upper and lower limits and get the solution of the given function volume and rotation under a bounded area.

You can see the below example in which you get an idea about shell method calculation.

## Solved Example of Shell Method:

A solved **example of shell method** is given below to let you know how the shell method calculator works.

**Example:**

Determine the volume of the solid formed by rotating the region bounded by the following about the y-axis. $$ y \;=\; 0,\; y \;=\; \frac{1}{(1+x^2)},\; x \;=\; 0 \;and\; x \;=\; 1 $$

**Solution:**

The given data is r(x) = x, h(x) = 1/(1 + x^{2}), x = 0, x = 1. Add the values in the formula,

$$ r(x) \;=\; x, h(x) \;=\; f(x) - g(x) $$

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

$$ V \;=\; 2 \pi \int_0^1 \frac{x}{1 + x^2} dx $$

Integrate the above function with respect to x. For integration, use the U substitution method,

$$ u \;=\; 1 + x^2, \;so\; du \;=\; 2x\; dx $$

The new limit is,

$$ u(0) \;=\; 1 \;and\; u(1) \;=\; 2 $$

Put these values in the above function,

$$ =\; \pi \int_1^2 \frac{1}{u} du $$

Integrate it with respect to u,

$$ =\; \pi ln\; u \biggr|_1^2 $$

Apply upper and lower limits, the solution is,

$$ =\; \pi ln\; 2 - \pi ln\; 1 $$

$$ =\; \pi ln\; 2 \approx 2.178\; units^3 $$

## How to Use the Cylindrical Shell Calculator?

Shell method formula calculator has a user-friendly layout that allows you to calculate shell method questions. Before adding the input function, there are some steps that you should follow. These steps are:

- Enter the function that you want to evaluate using the shell method in the input field.
- Add the upper and the lower limit for the given function in the input field.
- Add the variable in which you want to evaluate integration for the shell method.
- Recheck your input function value before hitting the calculate button of shell volume calculator to start the calculation process.
- Click on the “
**Calculate**” button to get the desired result of your given shell method questions with a solution. - If you are trying our shell method calculator for the first time then you use the load examples option to see the exact calculations.
- The “Recalculate” button will give you a new page for solving shell method problems.

## Final Result of Method of Cylindrical Shells Calculator:

Shell calculator gives you the **solution** to a given problem when you add the input value to it. It provides you with the solutions which may contain as:

**Result Option**:

When you click on the result option, you get the solution of shell method question.

**Possible Step**:

When you click on the possible steps option it provides you with the solution of shell method function.

## Advantages of Shell Method Formula Calculator:

The shell volume calculator has many advantages that you obtain when you use it to calculate shell method problems. These advantages are:

- Cylindrical shell method calculator is an
**adaptable tool**that solves various types of shell method problems and give you solutions for the cross-sectional area. - It is a free tool so you can use it freely to find the shell method problems with a solution without spending anything.
- Our method of cylindrical shells calculator saves the time and effort that you consume in doing lengthy and complex calculations of shell method.
- It is an educational tool so you can use the cylindrical shell calculator for practice.
- It is a trustworthy tool that provides you with accurate solutions according to your input whenever you use this tool to get a solution.
- Shell method formula calculator provides you solutions with a complete process in a step-by-step method for a better understanding of Shell method problems.