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Average area of image of rotated square

Suppose we have a unit square that is rotating in the x (or y) axis. At a point in time, it stops rotating. The image of it viewed from directly on top of it should be a rectangle. What is the average area of this rectangle?

Note by Daniel Liu
4 years, 6 months ago

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If it rotates an angle of $$\theta$$ from the $$xy$$-plane, then the area (viewed from above) will be $$|\cos \theta|$$.

Therefore, the average area is $$\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi} |\cos \theta|\,d\theta = \dfrac{2}{\pi}$$.

- 4 years, 5 months ago

How do you mean, it should be a rectangle? If you have a square, then you can rotate it all you want, but it stays a square, right?

- 4 years, 5 months ago

But it is rotating in respect to the x axis, not the z axis.

- 4 years, 5 months ago

Can you rephrase it? I don't get how it's rotating...

- 4 years, 5 months ago

Hmm... I'm not really sure how to say it. Imagine a square. You stick a pole through two opposite midpoints of the square, and rotate it with that pole as the axis. That kind of rotating.

- 4 years, 5 months ago