Variation of functional with area and volume term

Question

Let $M$ be a closed manifold in $\mathbb{R}^3$ and $\partial M$ its surface. I want to find (in general terms) the mannifold that minimizes a functional of the form $$I[M]=\int_{\partial M}f\,\mathrm{d}S+\int_MF\,\mathrm{d}V,$$ for some functions $F,f$. Let $g_{ij}$ be the metric tensor of the mannifold and $g$ its determinant. The volume term is easily rewritten: $$\int_MF\,\mathrm{d}V=\int F(\vec{x})\sqrt{-g}\,\mathrm{d}^3x,$$ but what should I do with the surface term? The metric tensor is three-dimensional but for the surface integral I need only two dimensions. I've read at some point of induced metrics, they might be what I need. But then again, how do I vary the surface intergral in terms of an induced metric with respect to the actual metric?