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I was wondering if when we are setting up volume integrals, if we always integrate as 1? then $\delta r \delta \theta$ ? or do we use the function, and then if it ends up being 1, we go with that?

For example number 5 on the review sheet for the final exam. It says

"Let S denote the unit sphere. Evaluate $$\iiint_S (x^2 + y^2 + z^2) dV$$ "

So I was wondering if we set it up as...

$$ \int _0 ^{2\pi} \int _0 ^\pi \int _0^1 \rho^2* \rho^2 \sin \phi \delta \rho \delta \phi \delta \theta $$

or $$ \int _0 ^ {2\pi} \int _0 ^\pi \int _0^1 1 * \rho^2 \sin \phi \delta \rho \delta \phi \delta \theta $$

I was wondering if when we are setting up volume integrals, if we always integrate as 1? then $\delta r \rho \delta \phi \delta \theta$ ? or do we use the function, and then if it ends up being 1, we go with that?

For example number 5 on the review sheet for the final exam. It says

"Let S denote the unit sphere. Evaluate $$\iiint_S (x^2 + y^2 + z^2) dV$$ "

So I was wondering if we set it up as...

$$ \int _0 ^{2\pi} \int _0 ^\pi \int _0^1 \rho^2* \rho^2 \sin \phi \delta \rho \delta \phi \delta \theta $$

or $$ \int _0 ^ {2\pi} \int _0 ^\pi \int _0^1 1 * \rho^2 \sin \phi \delta \rho \delta \phi \delta \theta $$