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 + x2), 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.
