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Shell Method Calculator + Online Solver With Free Steps
The Shell Method Calculator is a helpful tool that determines the volume for various solids of revolution quickly. The calculator takes in the input details regarding the radius, height, and interval of the function.
If a two-dimensional region in a plane is rotated around a line in the same plane, it results in a three-dimensional object which is called a solid of revolution.
The volume of these objects can be determined by using integration like in the shell method.
The calculator outputs the numerical value of the volume of solid and indefinite integral for the function.
What Is a Shell Method Calculator?
A Shell Method Calculator is an online calculator made to quickly calculate the volume of any complex solid of revolution using the shell method.
Many real-life objects we observe are solid of revolution like revolving doors, lamps, etc. Such shapes are commonly used in the sector of mathematics, medicine, and engineering.
Therefore it is very important to find parameters like the surface area and volume of these shapes. Shell method is a common technique to determine the volume of solid of revolution. It involves integrating the product of radius and height of shape over the interval.
Finding the volume of the solid of revolution manually is a very tedious and time taking process. To solve it you require a strong grasp of mathematical concepts like integration.
But you can get relief from this rigorous process using Shell Method Calculator. This calculator is always accessible in your browser and it is very easy to understand. Just enter the required and get the most precise results.
How To Use the Shell Method Calculator?
You can use the Shell Method Calculator by entering equations for different solids of revolution in their respective boxes. The calculator’s front end contains four input boxes and one button.
To get optimal results from the calculator you must follow the below given detailed guidelines:
Step 1
First, enter the upper and lower limit of integral in the To and From boxes. These limits represent the interval of the variable.
Step 2
Then insert the equation for the height of the solid of revolution in the field Height. It will be a function of a variable either x or y which represents the height of a shape.
Step 3
Now put the value of radius in the Radius tab. It is the distance between the shape and the axis of rotation. It can be a numerical value or some value in terms of variables.
Step 4
At the last, click the Submit button for results.
Result
The solution to the problem is displayed in two portions. The first portion is the definite integral which gives the value of volume in numbers. Whereas the second portion is indefinite integral for the same function.
How Does the Shell Method Calculator Work?
This calculator works by finding the volume of solid of revolution via the shell method, which integrates the volume of solid over the bounded region. This is one of the most used applications of definite integrals.
There are different methods to calculate the volume of solids of revolution but before the discussion of methods, we should know about the solids of revolution first.
Solid of Revolution
The solid of revolution is a three-dimensional object obtained by rotating a function or a plane curve about a horizontal or vertical straight line that does not pass through the plane. This straight line is called an axis of revolution.
The definite integrals are used to find the volume of solid of revolution. Suppose that the solid is placed in the plane between the lines x=m and x=n. The cross-sectional area of this solid is A(x) which is perpendicular to the x-axis.
If this area is continuous on the interval [m,n], then the interval can be divided into several sub intervals of width $\Delta x$. The volume of all the sub-intervals can be found by summation of the volume of each sub-interval.
When the region is rotated about the x-axis which is bounded by the curve and the x-axis between the x=m and x=n then the volume formed can be calculated by the following integral:
\[V= \int_{m}^{n} A(x) \,dx\]
Similarly, when the region bounded by the curve and the y-axis between the y=u and y=v is rotated about the y-axis then the volume is given by:
\[V= \int_{u}^{v} A(y) \,dy\]
The volume of revolution has applications in geometry, engineering, and medical imaging. The knowledge of these volumes is also useful for manufacturing machine parts and creating MRI images.
There are different methods to find the volume of these solids which includes the shell method, disk method, and washer method.
The Shell Method
The shell method is the approach in which vertical slices are integrated over the bounded region. This method is proper where the vertical slices of the region can easily be considered.
This calculator also uses this method to find the volumes by decomposing the solid of revolution into cylindrical shells.
Consider the region in the plane that is split into several vertical slices. When any of the vertical slices will be rotated around the y-axis which is parallel to these slices, then a different object of revolution will be obtained which is called the cylindrical shell.
The volume of one individual shell can be obtained by multiplying the surface area of this shell by the thickness of the shell. This volume is given by:
\[\Delta V= 2 \pi xy\,\Delta x\]
Where 2 $\pi$ xy is the surface area of the cylindrical shell and $Delta x$ is the thickness or depth.
The volume of the whole solid of revolution can be calculated by summation of the volumes of each shell as the thickness goes to zero in the limit. Now the formal definition to calculate this volume is given below.
If a region R which is bounded by x=a and x=b is revolved around the vertical axis, then the solid of revolution is formed. The volume of this solid is given by following definite integral as:
\[V= 2\pi \int_{a}^{b} r(x) h(x) \,dx\]
Where r(x) is the distance from the axis of revolution, basically it is the radius of the cylindrical shell, and h is the height of the solid.
The integration in the shell method is along the coordinate axis which is perpendicular to the axis of rotation.
Special Cases
For the height and radius, there are the following two important cases.
- When the region R is bounded by y=f(x) and below by y=g(x), then the height h(x) of the solid is given by h(x)= f(x)-g(x).
- When the axis of revolution is the y-axis means that x=0, then r(x) = x.
When To Use the Shell Method
It is sometimes difficult to choose which method to use for calculating the volume of solid of revolution. However, some cases in which the shell method is more feasible to use are given below.
- When the function f(x) is revolved around a vertical axis.
- When the rotation is along the x-axis and the graph is not a function on x but it is the function on y.
- When the integration of $f(x)^2$ is difficult but the integration of xf(x) is easy.
Solved Example
To better understand the working of calculators, we need to go through some solved examples. Each example and its solution is explained briefly in the upcoming section.
Example 1
A student studying calculus is asked to find the volume of the solid of revolution formed by rotating the region bounded by $y= \frac{1}{1+x^2}$, x=0, and x=1 about the y-axis.
Solution
The volume of the solid can easily find out by inserting the required values in the Shell method calculator. This calculator solves the definite integral to calculate the required volume.
Definite Integral
\[2\pi \int_{0}^{1} \frac{1}{1+x^2} \,dx= 2.17759\]
Indefinite Integral
\[2\pi \int_{0}^{1} \frac{1}{1+x^2} \,dx= \pi\,\log(x^2+1) + constant\]
Example 2
An electrical engineer encountered a signal on an oscilloscope that has the following height and radius function.
Height, h(x) = $\sqrt {x}$
Radius, r(x) = x
He needs to find the volume of the shape if revolved around the y within the interval x = [0,4] to further determine the characteristics of the signal.
Solution
The above problem is solved by this superb calculator and the answer is as follows:
Definite Integral
\[ 2\pi \int_{0}^{4} x^{ \frac{3}{2} } \, dx = 80.2428 \]
Indefinite Integral
\[ 2\pi \int_{0}^{4} x^{ \frac{3}{2} } \, dx = \frac{4}{5} \pi x^{ \frac{5}{2} } + constant \]
Example 3
A mathematician is required to calculate the volume of solid of revolution made by rotating the shape around the y-axis with the given characteristics:
Height, h(x) = x – $x^{3}$
Radius, r(x) = x
The interval for the shape is between x=0 and x=1.
Solution
The volume of the solid of revolution can be obtained using the Shell Method Calculator.
Definite Integral
\[ 2\pi \int_{0}^{1} x(x-x^{3}) \,dx = \frac{4\pi}{15} \approx 0.83776 \]
Indefinite Integral
\[ 2\pi \int_{0}^{1} x(x-x^{3}) \,dx = 2\pi \left( \frac{x^{3}}{3} – \frac{x^{5}}{5} \right) + constant \]