Cut Ellipsoid Volume

Calculus Level 5

An ellipsoid is given by

\[ \dfrac{x^2}{100} + \dfrac{y^2}{225} + \dfrac{z^2}{900} = 1. \]

It is cut by the plane

\[ x + 3 y + 2 z = 40. \]

Find the volume of the region that is inside the ellipsoid and below the plane. It is assumed that the upward direction is along the positive \( z \)-axis. This 3D region is depicted in the figure above.

If the volume is \( V \), enter \( \lfloor V \rfloor \) as your answer.

Hint: Use scaling to transform the ellipsoid and the cutting plane into a sphere with a corresponding cutting plane.


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