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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