Area of An Integral Rotated Around The X Axis Calculator
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Reviewed by Calculator Editorial Team
Introduction
When a function is rotated around the x-axis, the area of the resulting solid can be calculated using the disk method. This method involves integrating the area of circular cross-sections perpendicular to the x-axis.
Key Formula
The area A of the solid formed by rotating the function y = f(x) around the x-axis from x = a to x = b is given by:
\[ A = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
The disk method is particularly useful for functions that are non-negative and continuous over the interval [a, b]. The result represents the volume of revolution in cubic units.
Disk Method
The disk method works by:
- Identifying the function y = f(x) to be rotated
- Determining the interval [a, b] over which to integrate
- Squaring the function to get the radius of each circular disk
- Multiplying by π to get the area of each disk
- Integrating the result to sum all the infinitesimal disks
Note: The function must be continuous and non-negative on the interval [a, b] for the disk method to be valid.
For more complex shapes, the shell method might be more appropriate, but the disk method provides a straightforward approach for many common functions.
Worked Example
Let's calculate the area of the solid formed by rotating y = √x around the x-axis from x = 0 to x = 4.
Example Calculation
\[ A = \pi \int_{0}^{4} (\sqrt{x})^2 \, dx = \pi \int_{0}^{4} x \, dx \]
\[ = \pi \left[ \frac{x^2}{2} \right]_{0}^{4} = \pi \left( \frac{16}{2} - 0 \right) = 8\pi \]
The result is 8π cubic units, which represents the volume of the solid formed by the rotation.
FAQ
What if the function is negative?
The disk method requires the function to be non-negative. For negative functions, you would need to adjust the limits of integration or use the shell method.
Can I use the disk method for vertical rotation?
No, the disk method is specifically for rotation around the x-axis. For rotation around the y-axis, you would use the shell method.
What units should I use?
The result will be in cubic units of the input units. For example, if x is in meters, the result will be in cubic meters.