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Shell Method Calculator
The shell method is a powerful technique for calculating the volume of solids of revolution in calculus. This calculator simplifies the process of applying the shell method to various problems.
How to Use This Calculator
Enter the function for the outer radius R(x)
Enter the function for the inner radius r(x) (if applicable)
Specify the axis of rotation (y-axis or x-axis)
Enter the lower and upper limits of integration
Click "Calculate" to get the result
Shell Method Formula
The general formula for the shell method is:
V
=
2
π
∫
a
b
x
[
R
(
x
)
−
r
(
x
)
]
d
x
V=2π∫
a
b
​
x[R(x)−r(x)]dx
Where:
V is the volume of the solid
x is the distance from the axis of rotation
R(x) is the outer radius function
r(x) is the inner radius function
a and b are the lower and upper limits of integration
When to Use the Shell Method
The shell method is ideal for calculating volumes when:
The solid is formed by rotating a region around a vertical axis (usually the y-axis)
The region is defined by functions in terms of x
Integration with respect to x is simpler than with respect to y
Examples
Example 1: Wine Bottle
Calculate the volume of a wine bottle formed by rotating the region between y = x² and y = 4 around the y-axis.
Solution:
Outer radius: R(x) = 2
Inner radius: r(x) = x
Limits: x = 0 to x = 2
V
=
2
π
∫
0
2
x
(
2
−
x
)
d
x
=
16
π
3
≈
16.76
cubic units
V=2π∫
0
2
​
x(2−x)dx=
3
16π
​
≈16.76 cubic units
Example 2: Hollow Cylinder
Find the volume of a hollow cylinder with outer radius 5 and inner radius 3, height 10.
Solution:
Outer radius: R(x) = 5
Inner radius: r(x) = 3
Limits: https://shellmethodcalculator.info/ y = 0 to y = 10
V
=
2
π
∫
0
10
x
(
5
−
3
)
d
x
=
160
π
≈
502.65
cubic units
V=2π∫
0
10
​
x(5−3)dx=160π≈502.65 cubic units
Advantages of the Shell Method
Simplifies calculations for certain types of solids
Avoids issues with vertical asymptotes
Often easier when the region is defined by functions of x