Finding suitable element to perform integration upon [closed]

Is there any precise (proper) method or technique to specify the element on which integration will be performed. Is it the same for all properties like moment of inertia, gravitational potential, gravitational field etc? How can one explain this?

For e.g. For finding different properties for objects such as ring, shell, sphere, cylinder, cone, lamina etc., we first find an elemental mass and then integrate it.

• It's not completely clear what you're asking. But I agree that analysing a system so that you can then apply integration can be tricky! – PM 2Ring Jul 6 '19 at 15:14
• Yes, in certain situations. I was looking for a methodology for solving this type of problem. – Shishir Maharana Jul 6 '19 at 15:16

• Find the volume of an homogeneous cylinder. Of course you can do so in the cartesian/spherical coordinate system, but using the cylindrical set $$(r,\phi, z)$$ is easier because the cylinder is obviously symmetrical in this system. Now, the smallest element you can "build" using this set is $$dr\cdot dz \cdot r d\phi$$, and integrating in this system is trivial. If you want to use another one, you just need to know what the Jacobian of the transformation is, and you're done.
• Find the volume of an homogeneous torus. Now this is trickier: there aren't any trivial simmetries nor in cartesian/spherical/cylindrical coordinate system. But it has some symmetries: it's polarly symmetrical around its axes, and you can use this to find its volume. Symmetry around the "internal" axis gives you a $$dr\cdot r d\phi$$, while symmetry around the "in the hole" axis gives you just a $$R d\varphi$$. Integrate those and you'll get the familiar $$2\pi^2Rr^2$$.