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A few years ago I was asked to solve an integration problem:

Let $D = \left\{ (x,y,z) \,|\, x^2+y^2+\frac{z^2}{4} \le 1 \right\}$. Find $\int_D z^2 dV$.

Of course, the problem was proposed for Jacobian. But an idea flashed into my mind. If we slice $D$ by planes perpendicular to $z$-axis, each thin circular slice has uniform density.

Note that a slice at $z$ has radius $\sqrt{ 1 - \frac{z^2}{4} }$ and thickness $dz$. So, the slice has mass $$ z^2 \pi \left( 1 - \frac{z^2}{4} \right) dz $$ and, thus, the total mass is $$ \int_D z^2 dV = 2\int_0^2 z^2 \pi \left( 1 - \frac{z^2}{4} \right) dz = \frac{32}{15}\pi.$$

I could answer by mental calculation and the proposer was surprised. LOL

It's very interesting to solve a mathematical problem by physical intuition.

What is your favorite problem which can be easily solved by physical intuition?

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Somewhat related to OP's example, Cavalieri's principle:

  • 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas.

  • 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes.

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  • $\begingroup$ Right. The idea in OP is actually Cavalieri's principle. But one of the proposers never understood the method. $\endgroup$
    – P.-S. Park
    Feb 17, 2017 at 8:50

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