# Surface Integrals over Basic Surfaces

Recall from the Surface Integrals page that if $w = f(x, y, z)$ is a three variable bounded real-valued function over the smooth finite area surface $\delta$ in $\mathbb{R}^3$ then the surface integral of $f$ over $\delta$ where $\delta$ is defined parametrically by $\vec{r}(u, v) = (x(u, v), y(u, v), z(u, v))$ for $(u, v) \in D$ is given by:

(1)

The most basic type of surface integral that we can evaluate are surfaces $\delta$ defined as a two variable real-valued function $z = g(x, y)$. We can parameterize the function $z = g(x, y)$ easily by letting $x = x$, $y = y$, and $z = g(x, y)$ (which we saw on the Parametric Surfaces page).

Now suppose that the first partial derivatives of $g$ are continuous on the domain $D$. Let’s compute the Jacobian determinants $\frac{\partial (y, z)}{\partial (x, y)}$, $\frac{\partial (z, x)}{\partial (x, y)}$ and $\frac{\partial (x, y)}{\partial (x, y)}$ required in our surface integral formula. We have that:

(2)

Therefore the surface integral of $w = f(x, y, z)$ over the surface $\delta$ where $\delta$ generated by $z = g(x, y)$ over $D$ on the $xy$-plane can be evaluated with the following formula:

(3)

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