Triple Integrals
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Triple Integrals
Given a function \(f:\mathbb{R}^3 \to \mathbb{R}\) defined on a solid \(S \subset \mathbb{R}^3\), we define the triple integral of \(f\) over the solid \(S\)
\[\iiint_S f(x,y,z)\;dV\]
Examples
Example 1
Calculate \[\iiint_S z\;dV\] where \(S\) is the solid bounded by the \(xy\)-plane and the paraboloid \(x^2 + y^2 + z = 4\):
Example 2
Calculate \[\iiint_S xy^2 + z\;dV\] where \(S\) is the solid bounded by the three coordinate planes and the plane \(2x + 2y + z = 2\):