# Integral over a rotating ellipsoid

• Silviu
In summary, we need to rotate the axes with the standard 2-D rotation transformation and integrate with respect to the rotated x^2 term to solve for the integral over an ellipsoid rotating around the z axis with an angular speed of ω. The result may include a mix of x^2, y^2, and xy terms, with the xy term potentially canceling out due to symmetry. This solution can be compared to the unrotated case, where the result was (4pi/15)a^3bc.
Silviu

## Homework Statement

Calculate ##\int x^2 dV## over an ellipsoid with semi-axes a, b and c along x, y and z. rotating around the z axis with an angular speed ##\omega##.

## The Attempt at a Solution

I managed to calculate this in the case when it is not rotating and I got ##\frac{4\pi}{15}a^3bc##. But I am not sure how to do it now I expected to get the same result as before times ##cos^2(\omega t)## but the result seems to be ##cos^2(\omega t) \int x^2 dV + sin^2(\omega t)\int y^2 dV##. Can someone help me with this? Thank you!

This problem is new to me too, but to start with, the ellipsoid apparently has the equation ## \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 ##. For rotation about the z-axis, the standard two dimensional rotation transformation for angle ## \theta=\omega t ## should apply. You rotate the axes with such a transformation by angle ## \theta ##, and you get the equation of the rotated ellipsoid in the form ## A(x')^2+B(y')^2 +C x'y'+z^2=1 ##. You then need to integrate ## \int (x')^2 \, dV ##. ## \\ ## Editing: Suggestion instead: Keep the ellipsoid fixed and rotate the ## x^2 ## term in the integral ## \int x^2 \, dV ## using the 2-D rotation transformation. The ## x^2 ## term will become a mix of ## x^2 ##, ## y^2 ## and ## xy ## terms. I think the ## xy ## integral over the ellipsoid will cancel from symmetry. Do you agree? And yes, I agree with your last statement in the OP. And since you solved it for the unrotated case, you should be able to write down the answer by inspection of the form you got for the unrotated case.

Last edited:
@Silviu See my edited post above.

## 1. What is an integral over a rotating ellipsoid?

An integral over a rotating ellipsoid is a mathematical calculation that involves finding the volume of a three-dimensional shape that is rotating about one of its axes. This type of integral is commonly used in physics and engineering to calculate properties of rotating objects.

## 2. How is the integral over a rotating ellipsoid different from a regular integral?

The integral over a rotating ellipsoid is different from a regular integral because it takes into account the rotation of the shape. This means that the limits of integration and the integrand will depend on the orientation of the ellipsoid with respect to the chosen axis of rotation.

## 3. What are the applications of the integral over a rotating ellipsoid?

The integral over a rotating ellipsoid has many applications in physics and engineering. It is commonly used to calculate the moment of inertia of rotating objects, which is important for understanding their rotational motion. It is also used in the analysis of satellite orbits, the design of rotating machinery, and in geodesy to model the shape of the Earth.

## 4. How is the integral over a rotating ellipsoid calculated?

The integral over a rotating ellipsoid is calculated using a specific formula that takes into account the shape, size, and orientation of the ellipsoid. This formula involves the use of special mathematical functions, such as the incomplete elliptic integral of the first kind. In some cases, the integral can also be approximated using numerical methods.

## 5. Are there any real-world examples of the integral over a rotating ellipsoid?

Yes, there are many real-world examples of the integral over a rotating ellipsoid. For instance, it is used in the design of gyroscope sensors, which are used in navigation systems. It is also used in the analysis of the Earth's rotation and its effects on satellite orbits. Additionally, the integral over a rotating ellipsoid is used in the study of fluid dynamics, as rotating ellipsoids can be used to model the shape of vortex rings.

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