Is $dV$ always equal to $dx \, dy \, dz$ in triple integrals?

Conditions

• The current setup uses rectangular/Cartesian coordinates.
• In other coordinate systems, such as polar-type coordinates mentioned verbally, $dV$ may take a different form.

Reasoning, step by step

1. Identify the coordinate system being used.
2. For Cartesian coordinates, use $dV = dx \, dy \, dz$.
3. For other coordinate systems, derive or look up the appropriate expression for $dV$.
4. Substitute the correct $dV$ into the triple integral.

Example

The speaker says $dV$ is the same thing as width times height times depth, and adds that $dV$ does not always have to be $dx \, dy \, dz$ if other coordinates are used.

Common misconceptions

• Confusing $dV$ with only the Cartesian product $dx \, dy \, dz$.
• Assuming the volume element is the same in all coordinate systems.

Watch the explanation

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